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Space 

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Geometry 


SPACE  AND  GEOMETRY 


IN  THE  LIGHT  OF 

PHYSIOLOGICAL,  PSYCHOLOGICAL, 
AND  PHYSICAL  INQUIRY 


ERNST  MACH 


FROM  THE  GERMAN  BY 
THOMAS  J.  MCCORMACK 


THE  OPEN  COURT  PUBLISHING  COMPANY 
LASALLE  ILLINOIS 


HATH- 
5TAT. 

LIBRARY 


COPYRIGHT  BY 

THE  OPEN  COURT  PUBLISHING  CO. 
CHICAGO,  U.  S.  A. 

1906 
All  rights  reserved. 

ISBN  0-87548- 177-9 
OC128109876543 

Printed  in  the  United  States  of  America 


CONTENTS. 

I. 

ON  PHYSIOLOGICAL,  AS  DISTINGUISHED  FROM 

GEOMETRICAL,  SPACE   5 

II. 

ON  THE  PSYCHOLOGY  AND  NATURAL  DEVEL- 
OPMENT OF  GEOMETRY 38 

III. 

SPACE  AND  GEOMETRY  FROM  THE  POINT  OF 

VIEW  OF  PHYSICAL  INQUIRY 94 


ON  PHYSIOLOGICAL,  AS  DISTINGUISHED 
FROM  GEOMETRICAL,  SPACE 

THE  SPACE  OF  VISION. 

The  sensible  space  of  our  immediate  perception, 
which  we  find  ready  at  hand  on  awakening  to  full 
consciousness,  is  considerably  different  from  geo- 
metrical space.  Our  geometrical  concepts  have 
been  reached  for  the  most  part  by  purposeful  experi- 
ence. The  space  of  the  Euclidean  geometry  is 
everywhere  and  in  all  directions  constituted  alike; 
it  is  unbounded  and  it  is  infinite  in  extent.  On  the 
other  hand,  the  space  of  sight,  or  "visual  space/'  as 
it  has  been  termed  by  Johannes  Miiller  and  Hering, 
is  found  to  be  neither  constituted  everywhere  and  in 
all  directions  alike,  nor  infinite  in  extent,  nor  un- 
bounded.1 The  facts  relating-  to  the  vision  of  forms, 
which  I  have  discussed  in  another  place,  show  that 
entirely  different  feelings  are  associated  with  "up- 
ness"  and  "downness,"  as  well  as  with  "nearness" 
and  "farness."  "Rightness"  and  "leftness"  are  like- 


1  These  terms  are  used  in  Riemann  's  sense. 

5 


6  SPACE    AND    GEOMETRY. 

wise  the  expression  of  different  feelings,  although  in 
this  case  the  similarity,  owing  to  considerations  of 
physiological  symmetry,1  is  greater.  The  unlikeness 
of  different  directions  finds  its  expression  in  the  phe- 
nomena of  physiological  similarity.  The  apparent 
augmentation  of  the  stones  at  the  entrance  to  a  tun- 
nel as  we  rapidly  approach  it  in  a  railway  train,  the 
shrinkage  of  the  same  objects  on  the  train's  emerg- 
ing from  the  tunnel,  are  exceptionally  distinct  cases 
only  of  the  fact  of  daily  experience  that  objects  in 
visual  space  cannot  be  moved  about  without  suffer- 
ing expansion  and  contraction, — so  that  the  space 
of  vision  resembles  in  this  respect  more  the  space  of 
the  metageometricians  than  it  does  the  space  of 
Euclid. 

Even  familiar  objects  at  rest  exhibit  the  same 
peculiarities.  A  long  cylindrical  glass  vessel  tipped 
over  the  face,  a  walking-stick  laid  endwise  against 
one  of  the  eyebrows,  appear  strikingly  conical  in 
shape.  The  space  of  our  vision  is  not  only  bound- 
ed, but  at  times  it  appears  to  have  even  very  nar- 
row boundaries.  It  has  been  shown  by  an  experi- 
ment of  Plateau  that  an  after-image  no  longer  suf- 
fers appreciable  diminution  when  projected  upon  a 
surface  the  distance  of  which  from  the  eye  exceeds 
thirty  meters.  All  ingenuous  people,  who  rely  on 
direct  perception,  like  the  astronomers  of  antiquity, 
see  the  heavens  approximately  as  a  sphere,  finite  in 


1  Analysis  of  the  Sensations,  1886.     English  trans.     Chicago, 
1897,  p.  49  et  seq. 


PHYSIOLOGICAL  AND  METRIC  SPACE  7 

extent.  In  fact,  the  oblateness  of  the  celestial  vault 
vertically,  —  a  phenomenon  with  which  even 
Ptolemy  was  acquainted,  and  which  Euler  has  dis- 
cussed in  modern  times, — is  proof  that  our  visual 
space  is  of  unequal  extent  even  in  different  direc- 
tions. Zoth  appears  to  have  found  a  physiological 
explanation  of  this  fact,  closely  related  to  the  con- 
jecture of  Ptolemy,  in  that  he  interprets  the  phe- 
nomenon as  due  to  the  elevation  of  the  line  of  sight 
with  respect  to  the  head.1  The  narrow  boundaries 
of  space  follow,  indeed,  directly  from  the  possi- 
bility of  panoramic  painting.  Finally,  let  us  observe 
that  visual  space  in  its  origin  is  in  nowise  metrical. 
The  localities,  the  distances,  etc.,  of  visual  space 
differ  only  in  quality,  not  in  quantity.  What  we 
term  visual  measurement  is  ultimately  the  upshot 
of  primitive  physical  and  metrical  experiences. 

THE  SPACE  OF  TOUCH. 

Likewise  the  skin,  which  is  a  closed  surface  of 
complicated  geometrical  form,  is  an  agency  of  spa- 
tial perception.  Not  only  do  we  distinguish  the 
quality  of  the  irritation,  but  by  some  sort  of  a 
supplementary  sensation  we  also  distinguish  its 
locality.  Now  this  supplementary  sensation  need 
only  differ  from  place  to  place  (the  difference  in- 


1  Zoth  'B  researches  have  recently  been  completed  by  F.  Hille- 
brand,  "Theorie  der  scheinbaren  Groesse  bei  binocularem 
Sehen"  (Denkschrift  der  Wiener  Akademie,  math.-naturw.  Cl. 
Bd.  72,  1906). 


8  SPACE    AND    GEOMETRY. 

creasing  with  the  distance  apart  of  the  spots  irri- 
tated) for  the  purely  biological  needs  of  the  organ- 
ism to  be  satisfied.  The  great  discrepancies  that  the 
space-sense  of  the  skin  presents  with  metrical  space 
have  been  investigated  by  E.  H.  Weber.1  The  dis- 
tance apart  at  which  the  two  points  of  a  pair  of 
dividers  are  distinctly  recognizable,  is  from  fifty  to 
sixty  times  less  on  the  tip  of  the  tongue  than  it  is 
on  the  middle  of  the  back.  At  different  parts  the 
skin  shows  great  divergencies  of  spatial  sensibility. 
A  pair  of  dividers  the  points  of  which  enclose  the 
upper  and  lower  lips,  appears  sensibly  to  shut  when 
moved  horizontally  towards  the  side  of  the  face 
(Fig.  i).  If  the  points  of  the  dividers  be  placed 
on  two  adjacent  finger-tips  and  thence  carried  over 
the  fingers,  the  palm  of  the  hand,  and  down  the  fore- 
arm, they  will  appear  at  the  latter  point  to  close 
completely  (Fig.  2).  (The  real  path  of  the  points 
is  dotted  in  the  figure;  the  apparent,  marked  by 
lines.)  The  forms  of  bodies  that  touch  the  skin 
are  indeed  distinguished;2  but  the  spatial  sense  of 
the  skin  is  nevertheless  greatly  inferior  to  that  of 
the  eye,  although  the  tip  of  the  tongue  will  recog- 


1<<Ueber  den  Eaumsinn  und  die  Empfindungskreise  in  der 
Haut  und  im  Auge"  (Berichte  der  Kg.  Sachs.  Gesellsch.  der 
Wissenschaften,  math.-naturw.  Cl.  1852,  p.  85  et  seq.). 

2  Care  must  be  taken  that  the  bodies  come  into  intimate  con- 
tact with  the  skin.  Various  objects  having  been  placed  in  my 
paralyzed  hand,  I  was  unable  to  recognize  some,  and  the  con- 
clusion was  formed  that  the  sensibility  of  the  skin  had  been 
impaired.  But  the  conclusion  was  erroneous;  for  immediately 
after  the  examination,  I  had  another  person  close  my  hand  and 
I  recognized  at  once  all  objects  put  in  it. 


PHYSIOLOGICAL  AND  METRIC  SPACE  9 

nize  the  circular  form  of  the  cross-section  of  a  tube 
2  mm.  in  diameter. 

The  space  of  the  skin  is  the  analogue  of  a  two- 
dimensional,  finite,  unbounded  and  closed  Rieman- 
nian  space.  Through  the  sensations  induced  by  the 

OUJ 


Figs.  1'and  2. 

movements  of  the  various  members  of  the  body 
(notably  the  arms,  the  hands,  and  the  fingers) 
something  analogous  to  a  third  dimension  is  super- 
posed. Gradually  we  are  led  to  the  interpretation 
of  this  system  of  sensations  by  the  simpler  and  more 
salient  relations  of  the  physical  world.  Thus  we 


IO  SPACE    AND    GEOMETRY. 

estimate  with  considerable  exactness  the  thickness 
of  a  plate  that  we  grasp  in  the  dark  with  the  fore- 
finger and  thumb  of  our  hand ;  and  we  may  do  the 
same  tolerably  well  also  by  touching  the  upper  sur- 
face with  the  finger  of  one  hand  and  the  lower  with 
the  finger  of  the  other.  Haptic  space,  or  the  space 
of  touch,  has  as  little  in  common  with  metric  space 
as  has  the  space  of  vision.  Like  the  latter,  it  also 
is  anisotropic  and  non-homogeneous.  The  cardinal 
directions  of  the  organism,  "forwards  and  back- 
wards," "upwards  and  downwards,"  "right  and 
left,"  are  in  both  physiological  spaces  alike  non- 
equivalent. 

SENSE  OF  SPACE  DEPENDENT  ON  BIOLOGICAL 
FUNCTION. 

The  fact  that  our  sense  of  space  is  not  developed 
at  points  where  it  can  have  no  biological  function, 
should  not  be  a  cause  of  special  astonishment  to  us. 
What  purpose  could  it  serve  to  be  informed  con- 
cerning the  location  of  internal  organs  over  the 
functions  of  which  we  have  no  control  ?  Thus,  our 
sense  of  space  does  not  extend  to  any  great  distance 
into  the  interior  of  the  nostrils.  We  cannot  tell 
whether  we  perceive  scents  introduced  by  one  of  a 
pair  of  pipettes,  at  the  right  or  at  the  left.  (E.  H. 
Weber,  loc.  dt.t  p.  126.)  On  the  other  hand,  tactual 
sensibility,  in  the  case  of  the  ear,  according  to 
Weber,  extends  as  far  as  the  tympanum,  and  enables 
us  to  determine  whether  the  louder  of  two  sound- 


PHYSIOLOGICAL  AND  METRIC  SPACE  II 

impressions  comes  from  the  right  or  the  left.  Rough 
information  as  to  the  locality  of  the  source  of  the 
sound  may  be  effected  in  this  manner;  but  it  is 
inadequate  for  exact  purposes. 

CORRESPONDENCE  OF  PHYSIOLOGICAL  AND  GEOMET- 
RIC SPACE. 

Physiological  space,  thus,  has  but  few  qualities 
in  common  with  geometric  space.  Both  spaces 
are  threefold  manifoldnesses.  To  every  point  of 
geometric  space,  A,  B,  C,  D,  corresponds  a  point  A', 
B' ,  C' ,  D'  of  physiological  space.  If  C  lies  between 
B  and  D,  then  also  will  C  lie  between  B'  and  D' . 
We  may  also  say  that  to  a  continuous  motion 
of  a  point  in  geometric  space  there  corresponds 
a  continuous  motion  of  a  co-ordinate  point  in  physi- 
ological space.  I  have  remarked  elsewhere  that 
this  continuity,  which  is  merely  a  convenient  fiction, 
need  not  in  the  case  of  either  space  be  an  actual 
continuity.  As  every  system  of  sensations,  so  also 
the  system  of  space-sensations,  is  finite, — a  fact 
which  cannot  astonish  us.  An  endless  series  of  sen- 
sational qualities  or  intensities  is  psychologically 
inconceivable.  The  other  properties  of  visual  space 
also  are  adapted  to  biological  conditions.  The  bio- 
logical needs  would  not  be  satisfied  with  the  pure 
relations  of  geometric  space.  "Rightness,"  "left- 
ness,"  "aboveness,"  "belowness,"  "nearness,"  and 
"farness,"  must  be  distinguished  by  a  sensational 
quality.  The  locality  of  an  object,  and  not  merely 


12  SPACE    AND    GEOMETRY. 

its  relation  to  other  localities,  must  be  known,  if  an 
animal  is  to  profit  by  such  knowledge.  It  is  also 
advantageous  that  the  sensational  indices  of  visual 
objects  which  are  near  by  and  consequently  more 
important  biologically,  are  sharply  graduated; 
whereas  with  the  limited  stock  of  indices  at  hand  in 
the  case  of  remote  and  less  important  objects  econ- 
omy is  practiced. 

A  TELEOLOGICAL  EXPLANATION. 

We  shall  now  develop  a  simple  general  con- 
sideration, which  is  again  essentially  of  a  teleologi- 
cal  nature.  Let  several  distinct  spots  on  the  skin 
of  a  frog  be  successively  irritated  by  drops  of  acid ; 
the  frog  will  respond  to  each  of  the  several  irrita- 
tions with  a  specific  movement  of  defense  corre- 
sponding to  the  spot  irritated.  Qualitatively  like 
stimuli  affecting  different  elementary  organs  and 
entering  by  different  paths  give  rise  to  processes 
which  are  propagated  back  to  the  environment  of 
the  animal  again  by  different  organs  along  different 
paths.  As  self-observation  shows,  we  not  only 
recognize  the  sameness  of  the  irritational  quality 
of  a  burn  at  whatever  sensitive  spot  it  may 
occur,  but  we  also  distinguish  the  spots  irri- 
tated; and  our  conscious  or  unconscious  move- 
ment for  protection  is  executed  accordingly. 
The  same  holds  true  for  itching,  tickling,  pressure 
on  the  skin,  etc.  We  may  be  permitted  to  assume, 
accordingly,  that  in  all  these  cases  there  is  resident 


PHYSIOLOGICAL  AND  METRIC  SPACE  13 

in  the  sensation,  which  qualitatively  is  the  same, 
some  differentiating  constituent  which  is  due  to  the 
specific  character  of  the  elementary  organ  or  spot 
irritated,  or,  as  Hering  would  say,  to  the  locality  of 
the  attention.  Conditions  resembling  those  which 
hold  for  the  skin  doubtless  also  obtain  for  the  ex- 
tended surface  of  any  sensory  organ;  although,  as 
in  the  case  of  the  retina,  the  facts  are  here  somewhat 
more  complicated.  Instead  of  movements  for  pro- 
tection or  flight,  may  appear  also,  conformably  with 
the  quality  of  the  irritation,  movements1  of  attack, 
the  form  of  which  is  also  determined  by  the  spot 
irritated.  The  snapping  reflex  of  the  frog,  which 
is  produced  optically,  and  the  picking  of  young 
chicks,  may  serve  as  examples.  The  perfect  biologi- 
cal adaptation  of  large  groups  of  connected  elemen- 
tary organs  among  one  another  is  thus  very  dis- 
tinctly expressed  in  the  perception  of  space. 

ALL  SENSATION  SPATIAL  IN  CHARACTER. 

This  natural  and  ingenuous  view  leads  directly  to 
the  theory  advanced  by  Prof.  William  James,  ac- 
cording to  which  every  sensation  is  in  part  spatial 
in  character;  a  distinct  locality,  determined  by  the 
element  irritated,  being  its  invariable  accompani- 
ment. Since  generally  a  plurality  of  elements  en- 
ters into  play,  voluminousness  would  also  have  to 

*I  accept,  it  will  be  seen,  in  a  somewhat  modified  and  ex- 
tended form,  the  opinion  advanced  by  Wlassak.  Cf.  his  beau- 
tiful remarks,  "Ueber  die  statischen  Functionen  des  Ohrlaby- 
rinths,"  Vierteljahrssch.  f.  w.  Philos,  XVII.  1  s.  29. 


14  SPACE    AND    GEOMETRY. 

be  ascribed  to  sensations.  In  support  of  his  hypoth- 
esis James  frequently  refers  to  Hering.  This  con- 
ception is,  in  fact,  almost  universally  accepted  for 
optical,  tactual,  and  organic  sensations.  Many 
years  ago,  I  myself  characterized  the  relationship 
of  tones  of  different  pitch  as  spatial,  or  rather  as 
analogous  to  spatial;  and  I  believe  that  the  casual 
remark  of  Hering,  that  deep  tones  occupy  a  greater 
volume  than  high  tones,  is  quite  apposite.1  The 
highest  audible  notes  of  Koenig's  rods  give  as  a  fact 
the  impression  of  a  needle-thrust,  while  deep  tones 
appear  to  fill  the  entire  head.  The  possibility  of 
localizing  sources  of  sound,  although  not  absolute, 
also  points  to  a  relation  between  sensations  of  sound 
and  space.  In  the  first  place,  we  clearly  distinguish, 
in  the  case  of  high  tones  at  least,  whether  the  right 
or  the  left  ear  is  more  strongly  affected.  And 
although  the  parallel  between  binocular  vision  and 
binaural  audition,  which  Steinhauser2  assumes,  may 
possibly  not  extend  very  far,  there  exists,  neverthe- 
less, a  certain  analogy  between  them;  and  the  fact 
remains  that  the  localizing  of  sources  of  sound  is 
effected  preferentially  by  the  agency  of  high  tones3 


*I  am  unable  to  give  the  reference  for  this  remark  definitely; 
it  was  therefore  doubtless  made  to  me  orally.  Germs  of  a  sim- 
ilar view,  as  well  as  suggestions  toward  the  modern  physical 
theories  of  audition,  are  to  be  found  even  in  Johannes  Miiller 
(Zur  vergleich.  Physiolog.  des  Gesichtssinnes,  Leipsic,  1826,  p. 
455  et  seq.)- 

2  Steinhauser,   Ueber  binaureales  Horen.     Vienna.     1877. 

»"Ueber  die  Funktion  der  Ohrmuschel."  Troltsch,  Archiv 
fur  OhrenheilTcunde,  N.  F.,  Band  3,  S.  72. 


PHYSIOLOGICAL  AND  METRIC  SPACE  15 

(of  small  volume  and  more  sharply  distinguished 
locality). 

NON-COINCIDENCE  OF  THE  PHYSIOLOGICAL  SPACES. 

The  physiological  spaces  of  the  different  senses 
embrace  in  general  physical  domains  which  are  only 
in  part  coincident.  Almost  the  entire  surface  of 
the  skin  is  accessible  to  the  sense  of  touch,  but  only 
a  part  of  it  is  visible.  On  the  other  hand,  the 
sense  of  sight,  as  a  telescopic  sense,  extends  in 
general  very  much  farther  physically.  We  can- 
not see  our  internal  organs,  which,  like  the 
elementary  organs  of  sense,  we  feel  as  existing 
in  space  and  invest  with  locality  only  when 
their  equilibrium  is  disturbed;  and  these  same  or- 
gans fall  only  partly  within  range  of  the  sense  of 
touch.  Similarly,  the  determination  of  position  in 
space  by  means  of  the  ear  is  far  more  uncertain  and 
is  restricted  to  a  much  more  limited  field  than  that 
by  the  eye.  Yet,  loosely  connected  as  the  different 
space-sensations  of  the  different  senses  may  origin- 
ally have  been,  they  have  still  entered  into  connec- 
tion through  association,  and  that  system  which  has 
the  greater  practical  importance  at  the  time  being 
is  prepared  to  take  the  place  of  the  other  (James). 
The  space-sensations  of  the  different  senses  are  un- 
doubtedly related,  but  they  are  certainly  not  identi- 
cal. It  is  of  little  consequence  whether  all  these 
sensations  be  termed  space-sensations  or  whether 


l6  SPACE    AND    GEOMETRY. 

one  species  only  be  invested  with  this  name  and  the 
others  be  conceived  as  analogues  of  them. 

SENSATION  IN  ITS  BIOLOGICAL  RELATIONSHIP. 

If  sensation  generally,  inclusive  of  sensation  of 
space,  be  conceived  not  as  an  isolated  phenomenon, 
but  in  its  biological  functioning,  in  its  biological 
relationship,  the  entire  subject  will  be  rendered  more 
intelligible.  As  soon  as  an  organ  or  system  of  or- 
gans is  irritated,  the  appropriate  movements  are 
induced  as  reflexes.  If  in  complicated  biological 
conditions  these  movements  be  found  to  be  evoked 
spontaneously  in  response  to  a  part  only  of  the 
original  irritation,  in  response  to  some  slight  im- 
pulse, in  response  to  a  memory,  then  we  are 
obliged  to  assume  that  traces  corresponding  to  the 
character  of  the  irritation  as  well  as  to  that  of  the 
irritated  organs  must  be  left  behind  in  the  memory. 
It  is  intelligible  thus  that  every  sensory  field  has  its 
own  memory  and  its  own  spatial  order. 

The  physiological  spaces  are  multiple  manifold- 
nesses  of  sensation.  The  wealth  of  the  manifold- 
ness  must  correspond  to  the  wealth  of  the  elements 
irritated.  The  more  nearly  elements  of  the  same 
kind  lie  together,  the  more  nearly  are  they  akin 
embryologically,  and  the  more  nearly  alike  are  the 
space-sensations  which  they  produce.  If  A  and  B 
be  two  elementary  organs,  it  is  permissible  to  as- 
sume that  the  space-sensation  produced  by  each  of 
them  is  composed  of  two  constituent  parts,  a  and  b, 


PHYSIOLOGICAL  AND  METRIC  SPACE  VJ 

of  which  the  one,  af  diminishes  the  more,  and  the 
other,  b,  increases  the  more,  the  farther  B  is  re- 
moved from  A,  or  the  more  the  ontogenetic  rela- 
tionship of  B  to  A  decreases.  The  elements  situated 
in  the  series  AB  present  a  continuously  graduated 
onefold  manifoldness  of  sensation.  The  multiplicity 
of  the  spatial  manifoldness  must  be  determined  in 
each  case  by  a  special  investigation;  for  the  skin, 
which  is  a  closed  surface,  a  twofold  manifoldness 
would  suffice,  although  a  multiple  manifoldness  is 
not  excluded,  and  is,  by  reason  of  the  varying  im- 
portance of  different  parts  of  the  skin,  even  very 
probable. 

It  may  be  said  that  sensible  space  consists  of  a 
system  of  graduated  feelings  evoked  by  the  sensory 
organs,  which,  while  it  would  not  exist  without  the 
sense-impressions  arising  from  these  organs,  yet 
when  aroused  by  the  latter  constitutes  a  sort  of  scale 
in  which  our  sense-impressions  are  registered.  Al- 
though every  single  feeling  due  to  a  sensory  organ 
(feeling  of  space)  is  registered  according  to  its  spe- 
cific character  between  those  next  related  to  it,  a 
plurality  of  excited  organs  is  nevertheless  very  ad- 
vantageous for  distinctness  of  localization,  for  the 
reason  that  the  contrasts  between  the  feelings  of 
locality  are  enlivened  in  this  way.  Visual  space, 
therefore,  which  ordinarily  is  well  filled  with  ob- 
jects, thus  affords  the  best  means  of  localization. 
Localization  becomes  at  once  uncertain  and  fluctu- 
ant for  a  single  bright  spot  on  a  dark  background. 


1 8  SPACE  AND  GEOMETRY. 

ORIGIN  OF  THE  THREE  DIMENSIONS. 

It  may  be  assumed  that  the  system  of  space-sen- 
sations is  in  the  main  very  similar,  though  un- 
equally developed,  in  all  animals  which,  like  man, 
have  three  cardinal  directions  distinctly  marked  on 
their  bodies.  Above  and  below,  the  bodies  of  such 
animals  are  unlike,  as  they  are  also  in  front  and  be- 
hind and  to  the  right  and  to  the  left.  To  the  right 
and  the  left,  these  animals  are  apparently  alike,  but 
their  geometrical  and  mechanical  symmetry,  which 
subserves  purposes  of  rapid  locomotion,  should  not 
deceive  us  with  regard  to  their  anatomical  and  phy- 
siological asymmetry.  Though  the  latter  may  ap- 
pear slight,  it  is  yet  distinctly  marked  in  the  fact 
that  species  very  closely  allied  to  symmetrical  ani- 
mals sometimes  assume  strikingly  unsymmetrical 
forms.  The  asymmetry  of  the  plaice  (flatfish)  is  a 
familiar  instance,  while  the  externally  symmetric 
form  of  the  slug  forms  an  instructive  contrast  to 
the  unsymmetric  shapes  of  some  of  its  nearer  rela- 
tives. This  trinity  of  conspicuously  marked  cardi- 
nal directions  might  indeed  be  regarded  as  the  phy- 
siological basis  for  our  familiarity  with  the  three  di- 
mensions of  geometric  space. 

BIOLOGICAL  IMPORTANCE  OF  TACTUAL  SPACE. 

Visual  space  forms  the  clearest,  precisest,  and 
broadest  system  of  space-sensations;  but,  biologi- 
cally, tactual  space  is  perhaps  more  important.  Irri- 


PHYSIOLOGICAL  AND  METRIC  SPACE  IQ 

tations  of  the  skin  are  spatially  registered  from  the 
very  outset;  they  disengage  the  corresponding  pro- 
tective movement;  the  disengaged  movement  then 
again  induces  sensations  in  the  extended  or  con- 
tracted skin,  in  the  joints,  in  the  muscles,  etc.,  which 
are  associated  with  sensations  of  space.  The  first 
localizations  in  tactual  space  are  presumably  effected 
on  the  body  itself;  as  when  the  palm  of  the  hand, 
for  example,  is  carried  over  the  surface  of  the  thigh, 
which  also  is  sensitive  to  impressions  of  space.  In 
this  manner  are  experiences  in  the  field  of  tactual 
space  gathered.  But  the  attempt  which  is  frequent- 
ly made  of  deriving  tactual  space  psychologically 
from  such  experiences,  by  aid  of  the  concept  of  time 
and  on  the  assumption  of  spaceless  sensations,  is  an 
altogether  futile  one. 

VISUAL  AND  TACTUAL  SPACE  CORRELATED. 

It  is  my  opinion  that  the  space  of  touch  and  the 
space  of  vision  may  be  conceived  after  quite  the 
same  manner.  This  can  be  done  (so  far  as  I  can 
infer  from  what  has  already  been  attempted  in  this 
direction)  only  by  transferring  Bering's  view  of 
visual  space  to  tactual  space.  This  also  accords 
best  with  general  biological  considerations.  A 
newly-hatched  chick  notices  a  small  object,  looks 
toward  it,  and  immediately  pecks  at  it.  A  certain 
area  in  the  central  organ  is  excited  by  the  irritation, 
and  the  looking  movement  of  the  muscles  of  the  eye, 
as  well  as  the  picking  movements  of  the  head  and 


2O  SPACE    AND    GEOMETRY. 

neck,  are  forthwith  automatically  disengaged  there- 
by. The  excitation  of  the  above-mentioned  area  of 
the  central  organ,  which  on  the  one  hand  is  deter- 
mined by  the  geometric  locality  of  the  physical  irri- 
tation, is  on  the  other  hand  the  basis  of  the  space- 
sensation.  The  disengaged  muscular  movements 
themselves  become  a  source  of  sensations  in  greatly 
varying  degree.  Whereas  the  sensations  attending 
the  movements  of  the  eyes,  in  the  case  of  man  at 
least,  usually  disappear  almost  altogether,  the  move- 
ments of  the  muscles  made  in  the  performance  of 
work  leave  behind  them  a  powerful  impression. 
The  behavior  of  the  chick  is  quite  similar  to  that  of 
an  infant  which  spies  a  shining  object  and  snatches 
at  it. 

It  will  scarcely  be  questioned  that  in  addition  to 
optical  irritations  other  irritations,  acoustic,  ther- 
mal, and  gustatory  in  character,  are  also  able  to 
evoke  movements  of  prehension  or  defense,  espe- 
cially so  in  the  case  of  blind  people,  and  that  to  the 
same  movements,  the  same  irritated  parts  of  the  cen- 
tral organ,  and  therefore  also  the  same  sensation  of 
space,  will  correspond.  The  irritations  affecting 
blind  people  are,  as  a  general  thing,  merely  limited 
to  a  more  restricted  sphere  and  less  sharply  deter- 
mined as  to  locality.  The  system  of  spatial  sensa- 
tions of  such  people  must  at  first  be  rather  meager 
and  obscure;  consider,  for  instance,  the  situa- 
tion of  a  blind  person  endeavoring  to  protect  him- 
self from  a  wasp  buzzing  around  his  head.  Yet  edu- 


PHYSIOLOGICAL  AND  METRIC  SPACE  21 

cation  can  do  very  much  towards  perfecting  the 
spatial  sense  of  blind  people,  as  the  achievements  of 
the  blind  geometer  Saunderson  clearly  show.  Spar 
tial  orientation  must  notwithstanding  have  been 
somewhat  difficult  for  him,  as  is  proved  by  the  con- 
struction of  his  table,  which  was  divided  in  the  sim- 
plest manner  into  quadratic  spaces.  He  was  wont 
to  insert  pins  into  the  corners  and  centers  of  these 
squares  and  to  connect  their  heads  by  threads.  His 
highly  original  work,  however,  must  by  reason  of 
its  very  simplicity  have  been  particularly  easy  for 
beginners  to  understand;  thus  he  demonstrated  the 
proposition  that  the  volume  of  a  pyramid  is  equal 
to  one-third  of  the  volume  of  a  prism  of  the  same 
base  and  height  by  dividing  a  cube  into  six  congru- 
ent pyramids,  each  having  a  side  of  the  cube  for  its 
base  and  its  vertex  in  the  center  of  the  cube.1 

Tactual  space  exhibits  the  same  peculiarities  of 
anisotropy  and  of  dissimilarity  in  the  three  cardinal 
directions  as  visual  space,  and  differs  in  these  pe- 
culiarities also  from  the  geometric  space  of  Euclid. 
On  the  other  hand,  optical  and  tactual  space-sensa- 
tions are  at  many  points  in  accord.  If  I  stroke  with 
my  hand  a  stationary  surface  having  upon  it  dis- 
tinct tangible  objects,  I  shall  feel  these  objects  as  at 
rest,  just  as  I  should  feel  visual  objects  to  be  when 
voluntarily  causing  my  eyes  to  pass  over  them,  al- 
though the  images  themselves  actually  move  across 
the  retina.  On  the  other  hand,  a  moving  object 


1  Diderot,  Lettre  sur  les  aveugles. 


22  SPACE    AND    GEOMETRY. 

appears  in  motion  to  the  seeing  or  touching  organ 
either  when  the  latter  is  at  rest  or  when  it  is  follow- 
ing the  object.  Physiological  symmetry  and  simi- 
larity find  the  same  expression  in  the  two  domains, 
as  has  been  elsewhere  shown  in  detail  j1  but,  however 
intimately  allied  they  may  be,  the  two  systems  of 
space-sensations  cannot  nevertheless  be  identical. 
When  an  object  excites  me  in  one  case  to  look  at  it 
and  in  another  to  grasp  it,  certainly  the  portions  of 
the  central  organ  which  are  affected  must  be  in  part 
different,  no  matter  how  nearly  contiguous  they 
may  be.  If  both  results  take  place,  the  domain  is 
naturally  larger.  For  biological  reasons,  we  may 
expect  that  the  two  systems  readily  coalesce  by  asso- 
ciation, and  readily  adapt  themselves  to  one  another, 
as  is  actually  the  case. 

FEELINGS  OF  SPACE  INVOLVE  STIMULUS  TO 
MOTION. 

But  the  province  of  the  phenomena  with  which 
we  are  concerned  is  not  yet  exhausted.  A  chick  can 
look  at  an  object,  pick  at  it,  or  even  be  determined 
by  the  stimulus  presented  to  run  to  it,  turn  towards 
or  around  to  it.  A  child  that  is  creeping  toward  an 
objective  point,  and  then  some  day  gets  up  and  runs 
with  several  steps  toward  it,  acts  likewise.  We  are 
under  the  necessity  of  conceiving  these  cases,  which 
pass  continuously  into  one  another,  from  some  simi- 
lar point  of  view.  There  must  be  certain  parts  of 

1 Analysis  of  the  Sensations,  Eng.  trans.,  p.  50  et  seq. 


PHYSIOLOGICAL  AND  METRIC  SPACE  23 

the  brain  which,  having  been  irritated  in  a  compara- 
tively simple  manner,  on  the  one  hand  give  rise  to 
feelings  of  space  and  on  the  other  hand,  by  their 
organization,  produce  automatic  movements  which 
at  times  may  be  quite  complicated.  The  stimulus 
to  extensive  locomotion  and  change  of  orientation 
not  only  proceeds  from  optical  excitations,  but  may 
also  be  induced,  even  in  the  case  of  blind  animals, 
by  chemical,  thermal,  acoustic,  and  galvanic  excita- 
tions.1 In  point  of  fact,  we  also  observe  extensive 
movements  of  locomotion  and  orientation  in  animals 
that  are  constitutionally  blind  (blind  worms),  as 
well  as  in  such  as  are  blind  by  retrogression  (moles 
and  cave  animals).  We  may  accordingly  conceive 
sensations  of  space  as  determined  in  a  perfectly 
analogous  manner  both  in  animals  with  and  in  ani- 
mals without  sight. 

A  person  watching  a  centipede  creeping  uniform- 
ly along  is  irresistibly  impressed  with  the  idea  that 
there  proceeds  from  some  organ  of  the  animal  a 
uniform  stream  of  stimulation  which  is  answered 
by  the  motor  organs  of  its  successive  segments  with 
rhythmic  automatic  movements.  Owing  to  the  dif- 
ference of  phase  of  the  hind  as  compared  with  the 
fore  segments,  there  is  produced  a  longitudinal  wave 
which  we  see  propagated  through  the  legs  of  the 
animal  with  mechanical  regularity.  Analogous 
phenomena  cannot  be  wanting  in  the  higher  ani- 


*Loeb,  Vergleichende  Gehirnphysiologie,  Leipzig,  1899,  page 
108  et  seq. 


24  SPACE    AND    GEOMETRY. 

mals,  and  as  a  matter  of  fact  do  exist  there.  We 
have  an  analogous  case  during  active  or  passive  ro- 
tation about  the  vertical  axis,  when  the  irritation 
induced  in  the  labyrinth  disengages  the  well  known 
nystagmic  movements  of  the  eyes.  The  organism 
adapts  itself  so  perfectly  to  certain  regular  altera- 
tions of  excitations  that  on  the  cessation  of  these 
alterations  under  certain  circumstances  negative 
after-images  are  produced.  I  have  but  to  recall  to 
the  reader's  mind  the  experiment  of  Plateau  and  Op- 
pel  with  the  expanding  spiral,  which  when  brought 
to  rest  appears  to  shrink,  and  the  corresponding  re- 
sults which  Dvorak  produced  by  alterations  of  the 
intensity  of  light.  Phenomena  of  this  kind  led  me 
long  ago  to  the  assumption  that  there  corresponded 
to  an  alteration  of  the  stimulus  u  with  the  time  t, 

to  a  rate  of  alteration,     -^     a  special  process  which 

under  certain  circumstances  might  be  felt  and  which 
is  of  course  associated  with  some  definite  organ. 
Thus,  rate  of  motion,  within  the  limits  within 
which  the  perceiving  organ  can  adapt  itself,  is  felt  di- 
rectly; this  is  therefore  not  only  an  abstract  idea,  as 
is  the  speed  of  the  hand  of  a  clock  or  of  a  projectile, 
but  it  is  also  a  specific  sensation,  and  furnished  the 
original  impulse  to  the  formation  of  the  idea.  Thus, 
a  person  feels  in  the  case  of  a  line  not  only  a  succes- 
sion of  points  varying  in  position,  but  also  the  di- 
rection and  the  curvature  of  the  line.  If  the  inten- 
sity of  illumination  of  a  surface  is  given  by  w  = 


PHYSIOLOGICAL  AND  METRIC  SPACE  25 

du   du 

-3-^.   -j4  find  their  expression  in  sensation, — a  cir- 
dx*    dy* 

cumstance  which  points  to  a  complicated  relation- 
ship between  the  elementary  organs. 

THE  CENTRAL  MOTOR  ORGAN  AND  THE  WILL  TO 
MOVE. 

If  there  actually  exists,  then,  as  in  the  centipede, 
an  organ  which  on  simple  irritation  disengages  the 
complicated  movements  belonging  to  a  definite  kind 
of  locomotion,  it  will  be  permissible  to  regard  this 
simple  irritation,  provided  it  is  conscious,  as  the  will 
or  the  attention  appurtenant  to  this  locomotion  and 
carrying  the  latter  spontaneously  with  it.  At  the 
same  time,  it  will  be  recognized  as  a  need  of  the  or- 
ganism that  the  effect  of  the  locomotion  should  be 
felt  in  a  correspondingly  simple  manner. 

BIOLOGICAL  NECESSITY  PARAMOUNT. 

For  detailed  illustration,  we  will  revert  once  more 
to  the  consideration  of  visual  space.  The  perception 
of  space  proceeds  from  a  biological  need,  and  will 
be  best  understood  in  its  various  details  from  this 
point  of  view.  The  greater  distinctness  and  the 
greater  nicety  of  discrimination  exercised  at  a  sin- 
gle specific  spot  on  the  retina  of  vertebrate  animals 
is  an  economic  device.  By  it,  the  possibility  of  mov- 


26  SPACE    AND    GEOMETRY. 

ing  the  eye  in  response  to  changes  of  attention  is 
rendered  necessary,  but  at  the  same  time  the  dis- 
turbing effects  of  willed  movements  of  the  eyes  on 
the  sensations  of  space  induced  by  objects  at  rest 
have  to  be  excluded.  Perception  of  the  movement 
of  an  image  across  the  retina  when  the  retina  is  at 
rest,  perception  of  the  movement  of  an  object  when 
the  eye  is  at  rest,  is  a  biological  necessity.  As  for 
the  perception  of  objects  at  rest  in  the  unfrequent 
contingency  of  a  movement  of  the  eye  due  to  some 
occurrence  extrinsic  to  consciousness  (external  me- 
chanical pressure,  or  twitching  of  the  muscles),  this 
was  unnecessary  for  the  organism.  The  foregoing 
requirements  are  to  be  harmonized  only  on  the 
assumption  that  the  displacement  of  the  image  on 
the  retina  of  the  eye  in  voluntary  movement  is  off- 
set as  to  spatial  value  by  the  volitional  character  of 
the  movement.  It  follows  from  this  that  objects  at 
rest  may  be  made,  while  the  eye  also  is  at  rest,  to 
suffer  displacement  in  visual  space  by  the  tendency 
to  movement  merely,  as  has  been  actually  shown  by 
experiment.1  The  second  offsetting  factor  is  also 
directly  indicated  in  this  experiment.  The  organ- 
ism is  not  obliged,  further,  in  accomplishing  its 
adaptation,  to  take  account  of  the  second  contin- 
gency mentioned,  which  arises  only  under  patho- 
logical or  artificial  circumstances.  Paradoxical  as 
the  conditions  here  involved  may  appear,  and  far 
removed  as  we  may  still  be  from  a  causal  compre- 
hension of  them,  they  are  nevertheless  easily  under- 

1  Analysis  of  the  Sensations.     English  Trans.     Page  59. 


PHYSIOLOGICAL  AND  METRIC  SPACE  2*J 

stood  when  thus  viewed  ideologically  as  a  con- 
nected whole. 

SENSATIONS  OF  MOVEMENT. 

Shut  up  in  a  cylindrical  cabinet  rotating  about  a 
vertical  axis,  we  see  and  feel  ourselves  rotating, 
along  with  the  cylindrical  wall,  in  the  direction  in 
which  the  motion  takes  place.  The  impression 
made  by  this  sensation  is  at  first  blush  highly  para- 
doxical, inasmuch  as  there  exists  not  a  vestige  of  a 
reason  for  our  supposing  that  the  rotation  is  a  rela- 
tive one.  It  appears  as  if  it  would  be  actually  pos- 
sible for  us  to  have  sensations  of  movement  in  ab- 
solute space, — a  conception  to  which  no  physical 
significance  can  possibly  be  attached.  But  physio- 
logically the  case  easily  admits  of  explanation.  An 
excitation  is  produced  in  the  labyrinthine  canals  of 
the  internal  ear,1  and  this  excitation  disengages,  in- 
dependently of  consciousness,  a  reflex  rotary  move- 
ment of  the  eyes  in  a  direction  opposite  to  that  of 
the  motion,2  by  which  the  retinal  images  of  all  ob- 
jects resting  against  the  body  are  displaced  exactly 
as  if  they  were  rotating  in  the  direction  of  the  mo- 
tion. Fixing  the  eyes  intentionally  upon  some  such 
object,  the  rotation  does  not,  as  might  be  supposed, 
disappear.  The  eye's  tendency  to  motion  is  then  ex- 
actly counterbalanced  by  the  introduction  of  a  fac- 


1  Bewegungsempfindungen,  41  et  seq.     Leipsic,  1875. 

*  Breuer,  Vorlaufige  Mittheilung  im  Anzeiger  der  Ick.  Gesell- 
tchaft  der  Aerzte  in  Wien,  vom  tO.  Nov.  1873, 


28  SPACE    AND    GEOMETRY. 

tor  extrinsic  to  consciousness.1  We  have  here  the 
case  mentioned  above,  where  the  eye,  held  externally 
at  rest,  becomes  aware  of  a  displacement  in  the 
direction  of  its  tendency  to  motion.  But  what  be- 
fore appeared  as  a  paradoxical  exception  is  now  a 
natural  result  of  the  adaptation  of  the  organism, 
by  which  the  animal  perceives  the  motion  of  its 
own  body  when  external  objects  at  rest  remain  sta- 
tionary. Analogous  adaptive  results  with  which 
even  Purkynje  was  in  part  acquainted  are  met  with 
in  the  domain  of  the  tactile  sense.2 

The  eyes  of  an  observer  watching  the  water  rush- 
ing underneath  a  bridge  are  impelled  without  notice- 
able effort  to  follow  the  motion  of  the  flowing  water 
and  to  adapt  themselves  to  the  same.  If  the  ob- 
server will  now  look  at  the  bridge,  he  will  see  both 
the  latter  and  himself  moving  in  a  direction  oppo- 
site to  that  of  the  water.  Here  again  the  eye 
which  fixates  the  bridge  must  be  maintained  at  rest 
by  a  willed  motional  effort  made  in  opposition  to 
its  unconsciously  acquired  motional  tendency,  and 
it  now  sees  apparent  motions  to  which  no  real  mo- 
tions correspond. 

But  the  same  phenomena  which  appear  here  para- 
doxical and  singular  undoubtedly  serve  an  impor- 
tant function  in  the  case  of  progressive  motion  or 


1  Analysis  of  the  Sensations.    English  Trans.    Page  71. 

*  Purkynje,  ' '  Beitrage  zur  Kenntniss  des  Schwindels. ' '  M edi- 
zin.  Jahrbucher  des  osterreichischen  Staates,  VI.  Wien,  1820. 
"Versuehe  tiber  den  Schwindel,  10th  Bulletin  der  naturw.  Sec- 
tion der  schles.  Gesellschaft.  Breslau,  1825,  s.  25. 


PHYSIOLOGICAL  AND  METRIC  SPACE  2Q 

locomotion.  To  the  property  of  the  visual  apparatus 
referred  to  is  due  the  fact  that  an  animal  in  pro- 
gressive motion  sees  itself  moving  and  the  station- 
ary objects  in  its  environment  at  rest.1  Anomalies 
of  this  character,  where  a  body  appears  to  be  in 
motion  without  moving  from  the  spot  which  it 
occupies,  where  a  body  contracts  without  really 
growing  smaller  (which  we  are  in  the  habit  of 
calling  illusions  on  the  few  rare  occasions  when 
we  notice  them)  have  accordingly  their  important 
normal  and  common  function. 

As  the  process  which  we  term  the  will  to  turn 
round  or  move  forward  is  of  a  very  simple  nature, 
so  also  is  the  result  of  this  will  characterized  by 
feelings  of  a  very  simple  nature.  Fluent  spatial 
values  of  certain  objects,  instead  of  stable,  make 
their  appearance  in  the  domain  of  the  tactual  as  well 
as  the  visual  sense.  But  even  where  visual  and 
tactual  sensations  are  as  much  as  possible  excluded, 
unmistakable  sensations  of  motion  are  produced; 
for  example,  a  person  placed  in  a  darkened  room, 
with  closed  eyes,  on  a  seat  affording  support  to  the 
body  on  all  sides,  will  be  conscious  of  the  slightest 
progressive  or  angular  acceleration  in  the  move- 
ment of  his  body,  no  matter  how  noiselessly  and 
gently  the  same  may  be  produced.2  By  association, 
these  simple  sensations  also  are  translated  at  once 
into  the  motor  images  of  the  other  senses.  Between 

1  Analysis  of  the  Sensations.  English  Trans.  Pages  63,  64, 
71,  72. 

1  Bewegungsempfindungen,  Leipsic,  1875. 


30  SPACE    AND    GEOMETRY. 

this  initial  and  terminal  link  of  the  process  are  sit- 
uated the  various  sensations  of  the  extremities 
moved,  which  ordinarily  enter  consciousness,  how- 
ever, only  when  obstructions  intervene. 

PRIMARY  AND  SECONDARY  SPACE. 

We  have  now,  as  I  believe,  gained  a  fair  insight 
into  the  nature  of  sensations  of  space.  The  last- 
discussed  species  of  sensations  of  space,  which  were 
denominated  sensations  of  movement,  are  sharply 
distinguished  from  those  previously  investigated,  by 
their  uniformity  and  inexhaustibility.  These  sen- 
sations of  movement  make  their  appearance  only  in 
animals  that  are  free  to  move  about,  whereas  ani- 
mals that  are  confined  to  a  single  spot  are  restricted 
to  the  sensations  of  space  first  considered,  which 
we  shall  designate  primary  sensations  of  space,  as 
distinguished  from  secondary  sensations  (of  move- 
ment). A  fixed  animal  possesses  necessarily  a 
bounded  space.  Whether  that  space  be  symmetrical 
or  unsymmetrical  depends  upon  the  conditions  of 
symmetry  of  its  own  body.  A  vertebrate  animal 
confined  to  a  single  spot  and  restricted  as  to  orien- 
tation could  construct  only  a  bounded  space  which 
would  be  dissimilar  above  and  below,  before  and 
behind,  and  accurately  speaking  also  to  the  right 
and  to  the  left,  and  which  consequently  would  pre- 
sent a  sort  of  analogy  with  the  physical  properties  of 
a  triclinic  crystal.  If  the  animal  acquired  the  power 
of  moving  freely  about,  it  would  obtain  in  this  way 


PHYSIOLOGICAL  AND  METRIC  SPACE  3! 

in  addition  an  infinite  physiological  space;    for  the 
sensations  of  movement  always  admit  of  being  pro- 
duced anew  when  not  prevented  by  accidental  ex- 
ternal  hindrances.     Untrammeled  orientation,  the 
interchangeability  of  every  orientation  with  every 
other,  invests  physiological  space  with  the  property 
of  equality  in  all  directions.     Progressive  motion 
and  the  possibility  of  orientation  in  any  direction 
together  render  space  identically  constituted  at  all 
places  and  in  all  directions.     Nevertheless,  we  may 
remark  at  this  juncture  that  the  foregoing  result 
has  not  been  obtained  through  the  operation    of 
physiological  factors  exclusively,  for  the  reason  that 
orientation  with  respect  to  the  vertical,  or  the  di- 
rection of  the  acceleration  of  gravity,  is  not  alto- 
gether optional  in  the  case  of  any  animal.     Marked 
disturbances  of  orientation  with  respect  to  the  ver- 
tical  make   themselves   most   strongly   felt   in   the 
higher  vertebrate  animals  by  their  physico-physio- 
logical  results,  by  which  they  are  restricted  as  re- 
gards both  duration  and  magnitude.    Primary  space 
cannot  be  absolutely  supplanted  by  secondary  space, 
for  the  reason  that  it  is  phylogenetically  and  onto- 
genetically  older  and  stronger.     If  primary  space 
decreases   in  significance  during  motion,   the  sen- 
sation of  movement  in  its  turn  immediately  vanishes 
when  the  motion  ceases,  as  does  every  sensation 
which  is  not  kept  alive  by  reviviscence  and  contrast. 
Primary  space  then  again  enters  upon  its  rights.   It 
is  doubtless  unnecessary  to  remark  that  physiological 


32  SPACE    AND    GEOMETRY. 

space  is  in  no  wise  concerned  with  metrical  rela- 
tions. 


BIOLOGICAL  THEORY  OF  SPATIAL  PERCEPTION. 

We  have  assumed  that  physiological  space  is  an 
adaptive  result  of  the  interaction  of  the  elementary 
organs,  which  are  constrained  to  live  together  and 
are  thus  absolutely  dependent  upon  co-operation, 
without  which  they  would  not  exist.  Of  cardinal 
and  greatest  importance  to  animals  are  the  parts 
of  their  own  body  and  their  relations  to  one  an- 
other; outward  bodies  come  into  consideration  only 
in  so  far  as  they  stand  in  some  way  in  relation 
to  the  parts  of  the  animal  body.  The  conditions 
here  involved  are  physiological  in  character, — 
which  does  not  exclude  the  fact  that  every  part  of 
the  body  continues  to  be  a  part  of  the  physical  world, 
and  so  subject  to  general  physical  laws,  as  is  most 
strikingly  shown  by  the  phenomena  which  take 
place  in  the  labyrinth  during  locomotion,  or  by  a 
change  of  orientation.  Geometric  space  embraces 
only  the  relations  of  physical  bodies  to  one  another, 
and  leaves  the  animal  body  in  this  connection  alto- 
gether out  of  account. 

We  are  aware  of  but  one  species  of  elements  of 
consciousness:  sensations.  In  our  perceptions  of 
space  we  are  dependent  on  sensations.  The  char- 
acter of  these  sensations,  and  the  organs  that  are 
in  operation  while  they  are  felt,  are  matters  that 
must  be  left  undecided. 


PHYSIOLOGICAL  AND  METRIC  SPACE  33 

The  view  on  which  the  preceding  reflections  are 
based  is  as  follows :  The  feeling  with  which  an  ele- 
mentary organ  is  affected  when  in  action,  depends 
partly  upon  the  character  (or  quality)  of  the  irri- 
tation; we  will  call  this  part  the  sense-impression. 
A  second  part  of  the  feeling,  on  the  other  hand, 
may  be  conceived  as  determined  by  the  individuality 
of  the  organ,  being  the  same  for  every  stimulus  and 
varying  only  from  organ  to  organ,  the  degree  of 
variation  being  inversely  proportional  to  the  onto- 
genetic  relationship.  This  portion  of  the  feeling 
may  be  called  the  space-sensation.  Space-sensation 
can  accordingly  be  produced  only  when  there  is 
some  irritation  of  elementary  organs;  and  every 
time  the  same  organ  or  the  same  complexus  of  or- 
gans is  irritated,  every  time  the  same  concatenation 
of  organs  is  aroused,  the  same  space-sensation  is 
evoked.  We  make  only  the  same  assumptions  here 
with  regard  to  the  elementary  organs  that  we  should 
deem  ourselves  quite  justified  in  making  with  re- 
spect to  isolated  individual  animals  of  the  same 
phylogenetic  descent  but  different  degrees  of  af- 
finity. 

The  prospect  is  here  opened  of  a  phylogenetic  and 
ontogenetic  understanding  of  spatial  perception ;  and 
after  the  conditions  of  the  case  have  been  once  thor- 
oughly elucidated,  a  physical  and  physiological  ex- 
planation seems  possible.  I  am  far  from  thinking 
that  the  explanation  here  offered  is  absolutely  ade- 
quate or  exhaustive  on  all  sides ;  but  I  am  convinced 
that  I  have  made  some  approach  to  the  truth  by  it. 


34  SPACE    AND    GEOMETRY. 

THE  A  PRIORI  THEORY  OF  SPACE. 

Kant  asserted  that  "one  could  never  picture  to 
oneself  that  space  did  not  actually  exist,  although 
one  might  quite  easily  imagine  that  there  were  no 
objects  in  space."  To-day,  scarcely  any  one  doubts 
that  sensations  of  objects  and  sensations  of  space 
can  enter  consciousness  only  in  combination  with  one 
another;  and  that,  vice  versa,  they  can  leave  con- 
sciousness only  in  combination  with  one  another. 
And  the  same  must  hold  true  with  regard  to  the  con- 
cepts which  correspond  to  these  sensations.  If  for 
Kant  space  is  not  a  "concept,"  but  a  "pure  (mere?) 
intuition  a  priori,"  modern  inquirers  on  the  other 
hand  are  inclined  to  regard  space  as  a  concept,  and 
in  addition  as  a  concept  which  has  been  derived 
from  experience.  We  cannot  intuite  our  system  of 
space-sensations  per  se:  but  we  may  neglect  sensa- 
tions of  objects  as  something  subsidiary;  and  if  we 
overlook  what  we  have  done,  the  notion  may  easily 
arise  that  we  are  actually  concerned  with  a  pure  in- 
tuition. If  our  sensations  of  space  are  independent 
of  the  quality  of  the  stimuli  which  go  to  produce 
them,  then  we  may  make  predications  concerning 
the  former  independently  of  external  or  physical 
experience.  It  is  the  imperishable  merit  of  Kant 
to  have  called  attention  to  this  point.  But  this  basis 
is  unquestionably  inadequate  to  the  complete  devel- 
opment of  a  geometry,  inasmuch  as  concepts,  and  in 
addition  thereto  concepts  derived  from  experience, 
are  also  requisite  to  this  purpose. 


PHYSIOLOGICAL  AND  METRIC  SPACE  35 

PHYSIOLOGICAL  INFLUENCES  IN  GEOMETRY. 

Physiological,  and  particularly  visual,  space  ap- 
pears as  a  distortion  of  geometrical  space  when  de- 
rived from  the  metrical  data  of  geometrical  space. 
But  the  properties  of  continuity  and  threefold  mani- 
foldness  are  preserved  in  such  a  transformation, 
and  all  the  consequences  of  these  properties  may  be 
derived  without  recourse  to  physical  experience,  by 
our  representative  powers  solely. 

Since  physiological  space,  as  a  system  of  sensa- 
tions, is  much  nearer  at  hand  than  the  geometric 
concepts  that  are  based  thereon,  the  properties  of 
physiological  space  will  be  found  to  assert  them- 
selves quite  frequently  in  our  dealings  with  geo- 
metric space.  We  distinguish  near  and  remote 
points  in  our  figures,  those  at  the  right  from  those 
at  the  left,  those  at  the  top  from  those  at  the  bot- 
tom, entirely  by  physiological  considerations  and 
despite  the  fact  that  geometric  space  is  not  cog- 
nizant of  any  relation  to  our  body,  but  only  of  re- 
lations of  the  points  to  one  another.  Among  geo- 
metric figures,  the  straight  line  and  the  plane  are 
especially  marked 'out  by  their  physiological  prop- 
erties; as  they  are  indeed  the  first  objects  of  geo- 
metrical investigation.  Symmetry  is  also  distinctly 
revealed  by  its  physiological  properties,  and  attracts 
thus  immediately  the  attention  of  the  geometer. 
It  has  doubtless  also  been  efficacious  in  determining 
the  division  of  space  into  right  angles.  The  fact 
that  similitude  was  investigated  previously  to  other 


36  SPACE    AND    GEOMETRY. 

geometric  affinities  likewise  is  due  to  physiological 
facts.  The  Cartesian  geometry  of  co-ordinates  in  a 
manner  liberated  geometry  from  physiological  in- 
fluences, yet  vestiges  of  their  thrall  still  remain  in 
the  distinction  of  positive  and  negative  co-ordinates, 
according  as  these  are  reckoned  to  the  right  or  to 
the  left,  upward  or  downward,  and  so  on.  This  is 
convenient,  but  not  necessary.  A  fourth  co-ordinate 
plane,  or  the  determination  of  a  point  by  its  dis- 
tances from  four  fundamental  points  not  lying  in 
the  same  plane,  exempts  geometric  space  from  the 
necessity  of  constantly  recurring  to  physiological 
space.  The  necessity  of  such  restrictions  as 
"around  to  the  right"  and  "around  to  the  left,"  and 
the  distinction  of  symmetrical  figures  by  these 
means  would  then  be  eliminated.  The  historical  in- 
fluences of  physiological  space  on  the  development 
of  the  concepts  of  geometric  space  are,  of  course,  not 
to  be  eliminated. 

Also  in  other  provinces,  as  in  physics,  the  influ- 
ence of  the  properties  of  physiological  space  is 
traceable,  and  not  alone  in  geometry.  Even  sec^ 
ondary  physiological  space  is  considerably  different 
from  Euclidean  space,  owing  to  the  fact  that  the 
distinction  between  "above"  and  "below"  does  not 
absolutely  disappear  in  the  former.  Sosikles  of 
Corinth  (Herodotus  v.  92)  asseverated  that  "sooner 
should  the  heavens  be  beneath  the  earth  and  the 
earth  soar  in  the  air  above  the  heavens,  than  that 
the  Spartans  should  lose  their  freedom."  And  his 
assertion,  together  with  the  tirades  of  Lactantius 


PHYSIOLOGICAL  AND  METRIC  SPACE  37 

(De  falsa  sapientia,  c.  24)  and  St.  Augustine  (De 
civitate  deif  XVII.,  9),  against  the  doctrine  of  the 
antipodes,  against  men  hanging  with  inverted  heads 
and  trees  growing  downward,— considerations 
which  even  after  centuries  touch  in  us  a  sympathetic 
chord, — all  had  their  good  physiological  grounds. 
We  have,  in  fact  less  reason  to  be  astonished  at 
the  narrow-mindedness  of  these  opponents  of  the 
doctrine  of  the  antipodes  than  we  have  to  be  filled 
with  admiration  for  the  great  powers  of  abstrac- 
tion exhibited  by  Archytas  of  Tarentum  and  Aris- 
tarchus  of  Samos. 


ON    THE    PSYCHOLOGY   AND    NATURAL 
DEVELOPMENT  OF  GEOMETRY. 

For  the  animal  organism,  the  relations  of  the  dif- 
ferent parts  of  its  own  body  to  one  another,  and  of 
physical  objects  to  these  different  parts,  are  primarily 
of  the  greatest  importance.  Upon  these  relations  is 
based  its  system  of  physiological  sensations  of  space. 
More  complicated  conditions  of  life,  in  which  the 
simple  and  direct  satisfaction  of  needs  is  impossible, 
result  in  an  augmentation  of  intelligence.  The 
physical,  and  particularly  the  spatial,  behavior  of 
bodies  toward  one  another  may  then  acquire  a  medi- 
ate and  indirect  interest  far  transcending1  our  inter- 
est in  our  momentary  sensations.  In  this  way,  a 
spatial  image  of  the  world  is  created,  at  first  in- 
stinctively, then  in  the  practical  arts,  and  finally 
scientifically,  in  the  form  of  geometry.  The  mutual 
relations  of  bodies  are  geometrical  in  so  far  as  they 
are  determined  by  sensations  of  space,  or  find  their 
expression  in  such  sensations.  Just  as  without  sen- 
sations of  heat  there  would  have  been  no  theory  of 
heat,  so  also  without  sensations  of  space  th'ere 
would  be  no  geometry;  but  both  the  theory  of  heat 
and  the  theory  of  geometry  stand  additionally  in 
need  of  experiences  concerning  bodies;  that  is  to 
say,  both  must  pursue  their  inquiries  beyond  the 

88 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     39 

narrow  boundaries  of  the  domains  of  sense  that  con- 
stitute their  peculiar  foundation. 

THE  ROLE  OF  BODIES. 

Isolated  sensations  have  independent  significance 
only  in  the  lowest  stages  of  animal  life ;  as,  for  ex- 
ample, in  reflex  motions,  in  the  removal  of  some  dis- 
agreeable irritation  of  the  skin,  in  the  snapping  re- 
flex of  the  frog,  etc.  In  the  higher  stages,  attention 
is  directed,  not  to  space-sensation  alone,  but  to  those 
intricate  and  intimate  complexes  of  other  sensations 
with  space-sensations  which  we  call  bodies.  Bodies 
arouse  our  interest ;  they  are  the  objects  of  our  activi- 
ties. But  the  character  of  our  activities  is  coinci- 
dently  determined  by  the  place  of  the  body,  whether 
near  or  far,  whether  above  or  below,  etc., — in  other 
words,  by  the  space-sensations  characterizing  that 
body.  The  mode  of  reaction  is  thus  determined  by 
which  the  body  can  be  reached,  whether  by  extend- 
ing the  arms,  by  taking  few  or  many  steps,  by 
hurling  missiles,  or  what  not.  The  quantity  of  sen- 
sitive elements  which  a  body  excites,  the  number  of 
places  which  it  covers,  that  is  to  say,  the  volume  of 
the  body,  is,  all  other  things  being  the  same,  pro- 
portional to  its  capacity  for  satisfying  our  needs, 
and  possesses  a  consequent  biological  import.  Al- 
though our  sensations  of  sight  and  touch  are  pri- 
marily produced  only  by  the  surfaces  of  bodies,  nev- 
ertheless powerful  associations  impel  especially  prim- 
itive man  to  imagine  more,  or,  as  he  thinks,  to  per- 


4O  SPACE    AND    GEOMETRY 

ceive  more,  than  he  actually  observes.  He  imagines 
to  be  rilled  with  matter  the  places  enclosed  by  the 
surface  which  alone  he  perceives;  and  this  is  espe- 
cially the  case  when  he  sees  or  seizes  bodies  with 
which  he  is  in  some  measure  familiar.  It  requires 
considerable  power  of  abstraction  to  bring  to  con- 
sciousness the  fact  that  we  perceive  the  surfaces 
only  of  bodies, — a  power  which  cannot  be  ascribed 
to  primitive  man. 

Of  importance  in  this  regard  are  also  the  peculiar 
distinctive  shapes  of  objects  of  prey  and  utility. 
Certain  definite  forms,  that  is,  certain  specific  com- 
binations of  space-sensations,  which  man  learns  to 
know  through  intercourse  with  his  environment, 
are  unequivocally  characterized  even  by  purely  phy- 
siological features.  The  straight  line  and  the  plane 
are  distinguished  from  all  other  forms  by  their 
physiological  simplicity,  as  are  likewise  the  circle 
and  the  sphere.  The  affinity  of  symmetric  and 
geometrically  similar  forms  is  revealed  by  purely 
physiological  properties.  The  variety  of  shapes  with 
which  we  are  acquainted  from  our  physiological  ex- 
perience is  far  from  being  inconsiderable.  Finally, 
through  employment  with  bodily  objects,  physical 
experience  also  contributes  its  quota  of  wealth  to  the 
general  store. 

THE  NOTION  OF  CONSTANCY. 
Crude  physical  experience  impels  us  to  attribute 
to  bodies  a  certain  constancy.    Unless  there  are  spe- 
cial reasons  for  not  doing  so,  the  same  constancy  is 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     4! 

also  ascribed  to  the  individual  attributes  of  the  corn- 
plexus  "body"  j1  thus  we  also  regard  the  color,  hard- 
ness, shape,  etc.,  of  the  body  as  constant;  and  par- 
ticularly we  look  upon  the  body  as  constant  with 
respect  to  space,  as  indestructible.  This  assump- 
tion of  spatial  constancy,  of  spatial  substantiality, 
finds  its  direct  expression  in  geometry.  Our  physi- 
ological and  psychological  organization  is  independ- 
ently predisposed  to  emphasize  constancy;  for  gen- 
eral physical  constancies  must  necessarily  have  found 
lodgment  in  our  organization,  which  is  itself  phys- 
ical, while  in  the  adaptation  of  the  species  very  defi- 
nite physical  constancies  were  at  work.  Inasmuch 
as  memory  revives  the  images  of  bodies,  before  per- 
ceived, in  their  original  forms  and  dimensions,  it 
supplies  the  condition  for  the  recognition  of  the 
same  bodies,  thus  furnishing  the  first  foundation 
for  the  impression  of  constancy.  But  geometry  is 
additionally  in  need  of  certain  individual  experi- 
ences. 

Let  a  body  K  move  away  from  an  observer  A  by 
being  suddenly  transported  from  the  environment 
FGH  to  the  environment  MNO.  To  the  optical 
observer  A  the  body  K  decreases  in  size  and  assumes 
generally  a  different  form.  But  to  an  optical  ob- 
server B,  who  moves  along  with  K  and  who  always 
retains  the  same  position  with  respect  to  K,  K  re- 
mains unaltered.  An  analogous  sensation  is  ex- 
perienced by  the  tactual  observer,  although  the  per- 


See  my  Analysis  of  the  Sensations,  introductory  chapter. 


42  SPACE    AND    GEOMETRY 

spective  diminution  is  here  wanting  for  the  reason 
that  the  sense  of  touch  is  not  a  telepathic  sense.  The 
experiences  of  A  and  B  must  now  be  harmonized 
and  their  contradictions  eliminated, — a  requirement 
which  becomes  especially  imperative  when  the  same 
observer  plays  alternately  the  parts  of  A  and  of  B. 
And  the  only  method  by  which  they  can  be  har- 
monized is,  to  attribute  to  K  certain  constant  spatial 
properties  independently  of  its  position  with  respect 
to  other  bodies.  The  space-sensations  determined 
by  K  in  the  observer  A  are  recognized  as  dependent 
on  other  space-sensations  (the  position  of  K  with 
respect  to  the  body  of  the  observer  A).  But  these 
same  space-sensations  determined  by  K  in  A  are 
independent  of  other  space-sensations,  characteriz- 
ing the  position  of  K  with  respect  to  B,  or  with  re- 
spect to  FGH . . .  MNO.  In  this  independence  lies 
the  constancy  with  which  we  are  here  concerned. 
The  fundamental  assumption  of  geometry  thus 
reposes  on  an  experience,  although  on  one  of  an 
idealized  kind. 

THE  NOTION  OF  RIGIDITY. 

In  order  that  the  experience  in  question  may  as- 
sume palpable  and  perfectly  determinate  form,  the 
body  K  must  be  a  so-called  rigid  body.  If  the  space- 
sensations  associated  with  three  distinct  acts  of 
sense-perception  remain  unaltered,  then  the  condi- 
tion is  given  for  the  invariability  of  the  entire  corn- 
plexus  of  space-sensations  determined  by  a  rigid 
body.  This  determination  of  the  space-sensations 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     43 

produced  by  a  body  by  means  of  three  space-sensa- 
tional elements  accordingly  characterizes  the  rigid 
body,  from  the  point  of  view  of  the  physiology  of 
the  senses.  And  this  holds  good  for  both  the  visual 
and  the  tactual  sense.  In  so  doing  we  are  not  think- 
ing of  the  physical  conditions  of  rigidity  (in  de- 
fining which  we  should  be  compelled  to  enter  dif- 
ferent sensory  domains),  but  merely  of  the  fact 
given  to  our  spatial  sense.  Indeed,  we  are  now  re- 
garding every  body  as  rigid  which  possesses  the 
property  assigned,  even  liquids,  so  long  as  their 
parts  are  not  in  motion  with  respect  to  one  another. 

PHYSICAL  ORIGIN  OF  GEOMETRY. 
Correct  as  the  oft-repeated  asseveration  is  that 
geometry  is  concerned,  not  with  physical,  but  with 
ideal  objects,  it  nevertheless  cannot  be  doubted  that 
geometry  has  sprung  from  the  interest  centering  in 
the  spatial  relations  of  physical  bodies.  It  bears  the 
distinctest  marks  of  this  origin,  and  the  course  of  its 
development  is  fully  intelligible  only  on  a  considera- 
tion of  this  fact.  Our  knowledge  of  the  spatial 
behavior  of  bodies  is  based  upon  a  comparison  of 
the  space-sensations  produced  by  them.  With- 
out the  least  artificial  or  scientific  assistance  we  ac- 
quire abundant  experience  of  space.  We  can  judge 
approximately  whether  rigid  bodies  which  we  per- 
ceive alongside  one  another  in  different  positions  at 
different  distances,  will,  when  brought  successively 
into  the  same  position,  produce  approximately  the 
same  or  dissimilar  space-sensations.  We  know 


44  SPACE    AND    GEOMETRY 

fairly  well  whether  one  body  will  coincide  with 
another, — whether  a  pole  lying  flat  on  the  ground 
will  reach  to  a  certain  height.  Our  sensations  of 
space  are,  however,  subject  to  physiological  cir- 
cumstances, which  can  never  be  absolutely  identical 
for  the  members  compared.  In  every  case,  rigor- 
ously viewed,  a  memory-trace  of  a  sensation  is  nec- 
essarily compared  with  a  real  sensation.  If,  there- 
fore, it  is  a  question  of  the  exact  spatial  relation- 
ship of  bodies  to  one  another,  we  must  provide  char- 
acteristics that  depend  as  little  as  possible  on  physi- 
ological conditions,  which  are  so  difficult  to  control. 

MEASUREMENT. 

This  is  accomplished  by  comparing  bodies  with 
bodies.  Whether  a  body  A  coincides  with  another 
body  B,  whether  it  can  be  made  to  occupy  exactly 
the  space  filled  by  the  other — that  is,  whether  under 
like  circumstances  both  bodies  produce  the  same 
space-sensations — can  be  estimated  with  great  pre- 
cision. We  regard  such  bodies  as  spatially  or  geo- 
metrically equal  in  every  respect, — as  congruent. 
The  character  of  the  sensations  is  here  no  longer 
authoritative;  it  is  now  solely  a  question  of  their 
equality  or  inequality.  If  both  bodies  are  rigid 
bodies,  we  can  apply  to  the  second  body  B  all  the 
experiences  which  we  have  gathered  in  connection 
with  the  first,  more  convenient,  and  more  easily 
transportable,  standard  body  A.  We  shall  revert 
later  to  the  circumstance  that  it  is  neither  necessary 
nor  possible  to  employ  a  special  body  of  comparison, 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     45 

or  standard,  for  every  body.  The  most  convenient 
bodies  of  comparison,  though  applicable  only  after 
a  crude  fashion, — bodies  whose  invariance  during 
transportation  we  always  have  before  our  eyes, — 
are  our  hands  and  feet,  our  arms  and  legs.  The 
names  of  the  oldest  measures  show  distinctly  that 
originally  we  made  our  measurements  with  hands'- 
breadths,  forearms  (ells),  feet  (paces) ,  etc.  Noth- 
ing but  a  period  of  greater  exactitude  in  measure- 
ment began  with  the  introduction  of  conventional 
and  carefully  preserved  physical  standards;  the 
principle  remains  the  same.  The  measure  enables 
us  to  compare  bodies  which  are  difficult  to  move  or 
are  practically  immovable. 

THE  ROLE  OF  VOLUME. 

As  has  been  remarked,  it  is  not  the  spatial,  but 
predominantly  the  material,  properties  of  bodies  that 
possess  the  strongest  interest.  This  fact  certainly 
finds  expression  even  in  the  beginnings  of  geometry. 
The  volume  of  a  body  is  instinctively  taken  into  ac- 
count as  representing  the  quantity  of  its  material 
properties,  and  so  comes  to  form  an  object  of  con^ 
tcntion  long  before  its  geometric  properties  receive 
anything  approaching  to  profound  consideration.  It 
is  here,  however,  that  the  comparison,  the  measure- 
ment of  volumes  acquires  its  initial  import,  and 
thus  takes  its  place  among  the  first  and  most  im- 
portant problems  of  primitive  geometry. 

The  first  measurements  of  volume  were  doubtless 
of  liquids  and  fruit,  and  were  made  with  hollow 


46  SPACE    AND    GEOMETRY 

measures.  The  object  was  to  ascertain  conveniently 
the  quantity  of  like  matter,  or  the  quantity  (number) 
of  homogeneous,  similarly  shaped  (identical)  bod- 
ies. Thus,  conversely,  the  capacity  of  a  store-room 
(granary)  was  in  all  likelihood  originally  estimated 
by  the  quantity  or  number  of  homogeneous  bodies 
which  it  was  capable  of  containing.  The  measure- 
ment of  volume  by  a  unit  of  volume  is  in  all  prob- 
ability a  much  later  conception,  and  can  only  have 
developed  on  a  higher  stage  of  abstraction.  Esti- 
mates of  areas  were  also  doubtless  made  from  the 
number  of  fruit-bearing  or  useful  plants  which  a 
field  would  accommodate,  or  from  the  quantity  of 
seed  that  could  be  sown  on  it ;  or  possibly  also  from 
the  labor  which  such  work  required. 

MEASUREMENT  OF  SURFACES. 

The  measurement  of  a  surface  by  a  surface  was 
readily  and  obviously  suggested  in  this  connection 
when  fields  of  the  same  size  and  shape  lay  near  one 
another.  There  one  could  scarcely  doubt  that  the 
field  made  up  of  n  fields  of  the  same  size  and  form 
possessed  also  w-fold  agricultural  value.  We  shall 
not  be  inclined  to  underrate  the  significance  of  this 
intellectual  step  when  we  consider  the  errors  in  the 
measurement  of  areas  which  the  Egyptians1  and 
even  the  Roman  agrimensores2  commonly  com- 
mitted. 


1Eisenlohr,  Ein  mathematisches  Handbuch  der  alien 
ter:  Papyrus  Ehind,  Leipsic,  1877. 
aM.  Cantor,  Die  romischen  Agrimensoren,  Leipsic,  1875. 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     47 

Even  with  a  people  so  splendidly  endowed  with 
geometrical  talent  as  the  Greeks,  and  in  so  late  a 
period,  we  meet  with  the  sporadic  expression  of  the 
idea  that  surfaces  having  equal  perimeters  are 
equal  in  area.1  When  the  Persian  "Overman," 
Xerxes,2  wished  to  count  the  army  which  was  his  to 
destroy,  and  which  he  drove  under  the  lash  across 
the  Hellespont  against  the  Greeks,  he  adopted  the 
following  procedure.  Ten  thousand  men  were 
drawn  up  closely  packed  together.  The  area  which 
they  covered  was  surrounded  with  an  enclosure,  and 
each  successive  division  of  the  army,  or  rather,  each 
successive  herd  of  slaves,  that  was  driven  into  and 
filled  the  pen,  counted  for  another  ten  thousand.  We 
meet  here  with  the  converse  application  of  the  idea 
by  which  a  surface  is  measured  by  the  quantity 
(number)  of  equal,  identical,  immediately  adjacent 
bodies  which  cover  it.  In  abstracting,  first  instinc- 
tively and  then  consciously,  from  the  height  of  these 
bodies,  the  transition  is  made  to  measuring  surfaces 
by  means  of  a  unit  of  surface.  The  analogous  step 
to  measuring  volumes  by  volume  demands  a  far 
more  practiced,  geometrically  disciplined  intuition. 
It  is  effected  later,  and  is  even  at  this  day  less  easy 
to  the  masses. 

ALL  MEASUREMENT  BY  BODIES. 
The  oldest  estimates  of  long  distances,  which  were 
computed  by  days'  journeys,  hours  of  travel,  etc., 

1  Thucydides,  VI.,  1. 

8  Herodotus,  VII.,  22,  56,  103,  223. 


48  SPACE    AND    GEOMETRY 

were  based  doubtless  upon  the  effort,  labor,  and  ex- 
penditure of  time  necessary  for  covering  these  dis- 
tances. But  when  lengths  are  measured  by  the  re- 
peated application  of  the  hand,  the  foot,  the  arm, 
the  rod,  or  the  chain,  then,  accurately  viewed,  the 
measurement  is  made  by  the  enumeration  of  like 
bodies,  and  we  have  again  really  a  measurement 
by  volume.  The  singularity  of  this  conception  will 
disappear  in  the  course  of  this  exposition.  If,  now, 
we  abstract,  first  instinctively  and  then  consciously, 
from  the  two  transverse  dimensions  of  the  bodies 
employed  in  the  enumeration,  we  reach  the  meas- 
uring of  a  line  by  a  line. 

A  surface  is  commonly  defined  as  the  boundary 
of  a  space.  Thus,  the  surface  of  a  metal  sphere 
is  the  boundary  between  the  metal  and  the  air;  it 
is  not  part  either  of  the  metal  or  of  the  air;  two 
dimensions  only  are  ascribed  to  it.  Analogously, 
the  one-dimensional  line  is  the  boundary  of  a  sur- 
face; for  example,  the  equator  is  the  boundary  of 
the  surface  of  a  hemisphere.  The  dimensionless 
point  is  the  boundary  of  a  line;  for  example,  of  the 
arc  of  a  circle.  A  point,  by  its  motion,  generates  a 
one-dimensional  line,  a  line  a  two-dimensional  sur- 
face, and  a  surface  a  three-dimensional  solid  space. 
No  difficulties  are  presented  by  this  concept  to  minds 
at  all  skilled  in  abstraction.  It  suffers,  however, 
from  the  drawback  that  it  does  not  exhibit,  but  on 
the  contrary  artificially  conceals,  the  natural  and 
actual  way  in  which  the  abstractions  have  been 
reached.  A  certain  discomfort  is  therefore  felt 


PSYCHOLOGY   AND   DEVELOPMENT   OF    GEOMETRY     49 

when  the  attempt  is  made  from  this  point  of  view 
to  define  the  measure  of  surface  or  unit  of  area 
after  the  measurement  of  lengths  has  been  dis- 
cussed.1 

A  more  homogeneous  conception  is  reached  if 
every  measurement  be  regarded  as  a  counting  of 
space  by  means  of  immediately  adjacent,  spatially 
identical,  or  at  least  hypothetically  identical,  bodies, 
whether  we  be  concerned  with  volumes,  with  sur- 
faces, or  with  lines.  Surfaces  may  be  regarded  as 
corporeal  sheets,  having  everywhere  the  same  con- 
stant thickness  which  we  may  make  small  at  will, 
vanishingly  small;  lines,  as  strings  or  threads  of 
constant,  vanishingly  small  thickness.  A  point  then 
becomes  a  small  corporeal  space  from  the  extension 
of  which  we  purposely  abstract,  whether  it  be  part 
of  another  space,  of  a  surface,  or  of  a  line.  The 
bodies  employed  in  the  enumeration  may  be  of  any 
smallness  or  any  form  which  conforms  to  our  needs. 
Nothing  prevents  our  idealizing  in  the  usual  manner 
these  images,  reached  in  the  natural  way  indicated, 
by  simply  leaving  out  of  account  the  thickness  of 
the  sheets  and  the  threads. 

The  usual  and  somewhat  timid  mode  of  present- 
ing the  fundamental  notions  of  geometry  is  doubt- 
less due  to  the  fact  that  the  infinitesimal  method 
which  freed  mathematics  from  the  historical  and 
accidental  shackles  of  its  early  elementary  form,  did 


1  Holder,  Anschauung  und  DenTcen  in  der  Geometric,  Leipsic, 
1900,  p.  18.  W.  Killing,  Einfuhrung  in  die  Grundlagen  det 
Geometric,  Paderborn,  1898,  II.,  p.  22  et  seq. 


5O  SPACE    AND    GEOMETRY 

not  begin  to  influence  geometry  until  a  later  period 
of  development,  and  that  the  frank  and  natural 
alliance  of  geometry  with  the  physical  sciences  was 
not  restored  until  still  later,  through  Gauss.  But 
why  the  elements  shall  not  now  partake  of  the 
advantages  of  our  better  insight,  is  not  to  be  clearly 
seen.  Even  Leibnitz  adverted  to  the  fact  that  it 
would  be  more  rational  to  begin  with  the  solid  in 
our  geometrical  definitions.1 

METHOD  OF  INDIVISIBLES. 

The  measurement  of  spaces,  surfaces,  and  lines  by 
means  of  solids  is  a  conception  from  which  our  re- 
fined geometrical  methods  have  become  entirely 
estranged.  Yet  this  idea  is  not  merely  the  forerun- 
ner of  the  present  idealized  methods,  but  it  plays 
an  important  part  in  the  psychology  of  geometry, 
and  we  find  it  still  powerfully  active  at  a  late  period 
of  development  in  the  workshop  of  the  investigators 
and  inventors  in  this  domain. 

Cavalieri's  Method  of  Indivisibles  appears  best 
comprehensible  through  this  idea.  Taking  his  own 
illustration,  let  us  consider  the  surfaces  to  be  com- 
pared (the  quadratures)  as  covered  with  equidis- 
tant parallel  threads  of  any  number  we  will,  after 
the  manner  of  the  warp  of  woven  fabrics,  and  the 
spaces  to  be  compared  (the  cubatures)  as  filled  with 
parallel  sheets  of  paper.  The  total  length  of  the 


1  Letter  to  Vitale  Giordano,  Leibnizens  mathematische  Sctvrif- 
ten,  edited  by  Gerhardt,  Section  I.,  Vol.  I.,  page  199. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     5! 

threads  may  then  serve  as  measure  of  the  surfaces, 
and  the  total  area  of  the  sheets  as  measure  of  the 
volumes,  and  the  accuracy  of  the  measurement  may 
be  carried  to  any  point  we  wish.  The  number  of 
like  equidistant  bodies,  if  close  enough  together  and 
of  the  right  form,  can  just  as  well  furnish  the  nu- 
merical measures  of  surfaces  and  solid  spaces  as 
the  number  of  identical  bodies  absolutely  covering 
the  surfaces  or  absolutely  filling  the  spaces.  If  we 
cause  these  bodies  to  shrink  until  they  become  lines 
(straight  lines)  or  until  they  become  surfaces 
(planes),  we  shall  obtain  the  division  of  surfaces 
into  surface-elements  and  of  spaces  into  space-ele- 
ments, and  coincidently  the  customary  measure- 
ment of  surfaces  by  surfaces  and  of  spaces  by  spaces. 
Cavalieri's  defective  exposition,  which  was  not 
adapted  to  the  state  of  the  geometry  of  his  time,  has 
evoked  from  the  historians  of  geometry  some  very 
harsh  criticisms  of  his  beautiful  and  prolific  proce- 
dure.1 The  fact  that  a  Hclmholtz,  his  critical  judg- 
ment yielding  in  an  unguarded  moment  to  his  fancy, 
could,  in  his  great  youthful  work,2  regard  a  surface 
as  the  sum  of  the  lines  (ordinates)  contained  in  it, 
is  merely  proof  of  the  great  depth  to  which  this  orig- 
inal, natural  conception  reaches,  and  of  the  facility 
with  which  it  reasserts  itself. 


1  Weissenborn,  Principien  der  hoheren  Analysis  in  ihrer 
Entwiclcelung.  Halle,  1856.  Gerhardt,  Entdeckung  der  Ana- 
lysis. Halle,  1855,  p.  18.  Cantor,  Geschichte  der  Mathematik. 
Leipsic,  1892,  II.  Bd. 

1  Helmholtz,  Erhaltung  der  Kraft.     Berlin,  1847,  p.  14. 


52  SPACE    AND    GEOMETRY 

The  following  simple  illustration  of  Cavalieri's 
method  may  be  helpful  to  readers  not  thoroughly 
conversant  with  geometry.  Imagine  a  right  circu- 
lar cylinder  of  horizontal  base  cut  out  of  a  stack  of 
paper  sheets  resting  on  a  table  and  conceive  in- 
scribed in  the  cylinder  a  cone  of  the  same  base  and 
altitude.  While  the  sheets  cut  out  by  the  cylinder 
are  all  equal,  those  forming  the  cone  increase  in  size 
as  the  squares  of  their  distances  from  the  vertex. 
Now  from  elementary  geometry  we  know  that  the 


Fig.  3. 

volume  of  such  a  cone  is  one-third  that  of  the  cylin- 
der. This  result  may  be  applied  at  once  to  the  quadra- 
ture of  the  parabola  (Fig.  3).  Let  a  rectangle  be 
described  about  a  portion  of  a  parabola,  its  sides  co- 
inciding with  the  axis  and  the  tangent  to  the  curve 
at  the  origin.  Conceiving  the  rectangle  to  be  covered 
with  a  system  of  threads  running  parallel  to  y,  every 


PSYCHOLOGY   AND   DEVELOPMENT   OF    GEOMETRY     53 

thread  of  the  rectangle  will  be  divided  into  two  parts, 
of  which  that  lying  outside  the  parabola  is  propor- 
tional to  x*.  Therefore,  the  area  outside  the  para- 
bola is  to  the  total  area  of  the  rectangle  precisely  as 
is  the  volume  of  the  cone  to  that  of  the  cylinder, 
viz.,  as  i  is  to  3. 

It  is  significant  of  the  naturalness  of  Cavalieri's 
view  that  the  writer  of  these  lines,  hearing  of  the 
higher  geometry  when  a  student  at  the  Gymnasium, 
but  without  any  training  in  it,  lighted  on  very  simi- 
lar conceptions, — a  performance  not  attended  with 
any  difficulty  in  the  nineteenth  century.  By  the  aid 
of  these  he  made  a  number  of  little  discoveries,  which 
were  of  course  already  long  known,  found  Guldin's 
theorem,  calculated  some  of  Kepler's  solids  of  rota- 
tion, etc. 

PRACTICAL  ORIGIN  OF  GEOMETRY. 

We  have  then,  first,  the  general  experience  that 
movable  bodies  exist,  to  which,  in  spite  of  their  mo- 
bility, a  certain  spatial  constancy  in  the  sense  above 
described,  a  permanently  identical  property,  must  be 
attributed, — a  property  which  constitutes  the 
foundation  of  all  notions  of  measurement.  But  in 
addition  to  this  there  has  been  gathered  instinctively, 
in  the  pursuit  of  the  trades  and  the  arts,  a  consid- 
erable variety  of  special  experiences,  which  have 
contributed  their  share  to  the  development  of  geom- 
etry. Appearing  in  part  in  unexpected  form,  in 
part  harmonizing  with  one  another,  and  sometimes, 


54  SPACE    AND    GEOMETRY 

when  incautiously  applied,  even  becoming  involved 
in  what  appears  to  be  paradoxical  contradictions, 
these  experiences  disturb  the  course  of  thought  and 
incite  it  to  the  pursuit  of  the  orderly  logical  connec- 
tion of  these  experiences.  We  shall  now  devote  our 
attention  to  some  of  these  processes. 

Even  though  the  well  known  statement  of  Hero- 
dotus1 were  wanting,  in  which  he  ascribes  the  origin 
of  geometry  to  land-surveying  among  the  Egyp- 
tians ;  and  even  though  the  account  were  totally  lost2 
which  Eudemus  has  left  regarding  the  early  history 
of  geometry,  and  which  is  known  to  us  from  an  ex- 
tract in  Proclus,  it  would  be  impossible  for  us  to 
doubt  that  a  pre-scientific  period  of  geometry  ex- 
isted. The  first  geometrical  knowledge  was  ac- 
quired accidentally  and  without  design  by  way  of 
practical  experience,  and  in  connection  with  the  most 
varied  employments.  It  was  gained  at  a  time  when 
the  scientific  spirit,  or  interest  in  the  interconnection 
of  the  experiences  in  question,  was  but  little  devel- 
oped. This  is  plain  even  in  our  meager  history  of 
the  beginnings  of  geometry,  but  still  more  so  in  the 
history  of  primitive  civilization  at  large,  where  tech- 
nical geometrical  appliances  are  known  to  have  ex- 
isted at  so  early  and  barbaric  a  day  as  to  exclude 
absolutely  the  assumption  of  scientific  effort. 

All  savage  tribes  practice  the  art  of  weaving,  and 
here,  as  in  their  drawing,  painting,  and  wood-cut- 


*  James  Gow,  A  Short  History  of  Greek  Mathematics,  Cam- 
bridge, 1884,  p.  134. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     $5 

ting,  the  ornamental  themes  employed  consist  of  the 
simplest  geometrical  forms.  For  such  forms,  like 
the  drawings  of  our  children,  correspond  best  to 
their  simplified,  typical,  schematic  conception  of  the 
objects  which  they  are  desirous  of  representing  and 


Fig.  4. 

it  is  also  these  forms  that  are  most  easily  produced 
with  their  primitive  implements  and  lack  of  manual 
dexterity.  Such  an  ornament  consisting  of  a  series 
of  similarly  shaped  triangles  alternately  inverted,  or 
of  a  series  of  parallelograms  (Fig.  4),  clearly  sug- 
gests the  idea,  that  the  sum  of  the  three  angles  of  a 
triangle,  when  their  vertices  are  placed  together, 
makes  up  two  right  angles.  Also  this  fact  could 
not  possibly  have  escaped  the  clay  and  stone  work- 
ers of  Assyria,  Egypt,  Greece,  etc.,  in  constructing 
their  mosaics  and  pavements  from  differently  col- 
ored stones  of  the  same  shape.  The  theorem  of  the 
Pythagoreans  that  the  plane  space  about  a  point 
can  be  completely  filled  by  only  three  regular  poly- 
gons, viz.,  by  six  equilateral  triangles,  by  four 
squares,  and  by  three  regular  hexagons,  points  to 
the  same  source.1  A  like  origin  of  this  truth  is  re- 
vealed also  in  the  early  Greek  method  of  demonstrat- 

1  This  theorem  is  attributed  to  the  Pythagoreans  by  Proclus. 
Cf.  Govr,  A  Short  History  of  Greek  Mathematics,  p.  143,  foot- 
note. 


56  SPACE    AND    GEOMETRY 

ing  the  theorem  regarding  the  angle-sum  of  any 
triangle  by  dividing  it  (by  drawing  the  altitude) 
into  two  right-angled  triangles  and  completing  the 
rectangles  corresponding  to  the  parts  so  obtained.* 
The  same  experiences  arise  on  many  other  occa- 
sions. If  a  surveyor  walk  round  a  polygonal  piece 
of  land,  he  will  observe,  on  arriving  at  the  starting 


Fig.  5. 

point,  that  he  has  performed  a  complete  revolution, 
consisting  of  four  right  angles.  In  the  case  of  a 
triangle,  accordingly,  of  the  six  right  angles  con- 
stituting the  interior  and  exterior  angles  (Fig.  5) 
there  will  remain,  after  subtracting  the  three  ex- 
terior angles  of  revolution,  a,  b,  c,  two  right  angles 
as  the  sum  of  the  interior  angles.  This  deduction 
of  the  theorem  was  employed  by  Thibaut,2  a  con- 

1  Hankel,  Geschichte  der  MafhematiJc,  Leipsic,  1874,  p.  96. 

'Thibaut,  Grundriss  der  reinen  Mathematilc,  Gottingen,  1809, 
p.  177.  The  objections  which  may  be  raised  to  this  and  the 
following  deductions  will  be  considered  later.  [The  same  proof 
is  also  given  by  Playfair  (1813).  See  Halsted's  translation  of 
Bolyai's  Science  Absolute  of  Space,  p.  67. — Tr.~\ 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     57 

temporary  of  Gauss.  If  a  draughtsman  draw  a  tri- 
angle by  successively  turning  his  ruler  round  the  in- 
terior angles,  always  in  the  same  direction  (Fig.  6), 
he  will  find  on  reaching  the  first  side  again  that  if  the 
edge  of  his  ruler  lay  toward  the  outside  of  the  tri- 
angle on  starting,  it  will  now  lie  toward  the  inside. 
In  this  procedure  the  ruler  has  swept  out  the  in- 


Fig.  e. 

terior  angles  of  the  triangle  in  the  same  direction, 
and  in  doing  so  has  performed  half  a  revolution.1 
Tylor2  remarks  that  cloth  or  paper-folding  may  have 
led  to  the  same  results.  If  we  fold  a  triangular 
piece  of  paper  in  the  manner  shown  in  Fig.  7,  we 
shall  obtain  a  double  rectangle,  equal  in  area  to  one- 
half  the  triangle,  where  it  will  be  seen  that  the  sum 
of  the  angles  of  the  triangle  coinciding  at  a  is  two 

1  Noticed  by  the  writer  of  this  article  while  drawing. 

*  Tylor,  Anthropology,  An  Introduction  to  the  Study  of  Man, 
etc.,  German  trans.,  Brunswick,  1883,  p.  383. 


58  SPACE    AND    GEOMETRY 

right  angles.  Although  some  very  astonishing  re- 
sults may  be  obtained  by  paper-folding,1  it  can 
scarcely  be  assumed  that  these  processes  were  his- 
torically very  productive  for  geometry.  The  mate- 


Fig.  7. 

rial  is  of  too  limited  application,  and  artisans  em- 
ployed with  it  have  too  little  incentive  to  exact  ob- 
servation. 

EXPERIMENTAL  KNOWLEDGE  OF  GEOMETRY. 

The  knowledge  that  the  angle-sum  of  the  plane 
triangle  is  equal  to  a  determinate  quantity,  namely, 
to  two  right  angles,  has  thus  been  reached  by  ex- 
perience, not  otherwise  than  the  law  of  the  lever  or 
Boyle  and  Mariotte's  law  of  gases.  It  is  true  that 
neither  the  unaided  eye  nor  measurements  with  the 
most  delicate  instruments  can  demonstrate  abso- 
lutely that  the  sum  of  the  angles  of  a  plane  triangle 
is  exactly  equal  to  two  right  angles.  But  the  case 
is  precisely  the  same  with  the  law  of  the  lever  and 
with  Boyle's  law.  All  these  theorems  are  therefore 
idealized  and  schematized  experiences;  for  real 


*See,  for  example,  Sundara  Bow's  Geometric  Exercises  in 
Paper-Folding.  Chicago :  The  Open  Court  Publishing  Co.,  1901. 
—Jr. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     59 

measurements  will  always  show  slight  deviations 
from  them.  But  whereas  the  law  of  gases  has  been 
proved  by  further  experimentation  to  be  approxi- 
mate only  and  to  stand  in  need  of  modification  when 
the  facts  are  to  be  represented  with  great  exactness, 
the  law  of  the  lever  and  the  theorem  regarding  the 
angle-sum  of  a  triangle  have  remained  in  as  exact 
accord  with  the  facts  as  the  inevitable  errors  of  ex- 
perimenting would  lead  us  to  expect;  and  the  same 
statement  may  be  made  of  all  the  consequences  that 
have  been  based  on  these  two  laws  as  preliminary 
assumptions. 

Equal    and    similar   triangles   placed    in   paving 
alongside  one  another  with  their  bases  in  one  and 


Fig.  8. 


the  same  straight  line  must  also  have  led  to  a  very 
important  piece  of  geometrical  knowledge.  (Fig. 
8.)  If  a  triangle  be  displaced  in  a  plane  along  a 
straight  line  (without  rotation),  all  its  points,  in- 


60  SPACE    AND    GEOMETRY. 

eluding  those  of  its  bounding  lines,  will  describe 
equal  paths.  The  same  bounding  line  will  furnish, 
therefore,  in  any  two  different  positions,  a  system 
of  two  straight  lines  equally  distant  from  one  an- 
other at  all  points,  and  the  operation  coincidently 
vouches  for  the  equality  of  the  angles  made  by  the 
line  of  displacement  on  corresponding  sides  of  the 
two  straight  lines.  The  sum  of  the  interior  angles 
on  the  same  side  of  the  line  of  displacement  was 
consequently  determined  to  be  two  right  angles,  and 
thus  Euclid's  theorem  of  parallels  was  reached.  We 
may  add  that  the  possibility  of  extending  a  pave- 
ment of  this  kind  indefinitely,  necessarily  lent  in- 
creased obviousness  to  this  discovery.  The  sliding 
of  a  triangle  along  a  ruler  has  remained  to  this  day 
the  simplest  and  most  natural  method  of  drawing 
parallel  lines.  It  is  scarcely  necessary  to  remark 
that  the  theorem  of  parallels  and  the  theorem  of 
the  angle-sum  of  a  triangle  are  inseparably  con- 
nected and  represent  merely  different  aspects  of  the 
same  experience. 

The  stone  masons  above  referred  to  must  have 
readily  made  the  discovery  that  a  regular  hexagon 
can  be  composed  of  equilateral  triangles.  Thus  re- 
sulted immediately  the  simplest  instances  of  the 
division  of  a  circle  into  parts, — namely  its  division 
into  six  parts  by  the  radius,  its  division  into  three 
parts,  etc.  Every  carpenter  knows  instinctively  and 
almost  without  reflection  that  a  beam  of  rectangular 
symmetric  cross-section  may,  owing  to  the  perfect 
symmetry  of  the  circle,  be  cut  out  from  a  cylindrical 


PSYCHOLOGY   AND   DEVELOPMENT   OF    GEOMETRY     6l 

tree-trunk  in  an  infinite  number  of  different  ways. 
The  edges  of  the  beam  will  all  lie  in  the  cylindrical 
surface,  and  the  diagonals  of  a  section  will  pass 
through  the  center.  It  was  in  this  manner,  accord- 
ing to  Hankel1  and  Tylor,2  that  the  discovery  was 
probably  made  that  all  angles  inscribed  in  a  semi- 
circle are  right  angles. 

ROLE  OF  PHYSICAL  EXPERIENCES. 

A  stretched  thread  furnishes  the  distinguishing 
visualization*  of  the  straight  line.  The  straight  line 
is  characterized  by  its  physiological  simplicity.  All 
its  parts  induce  the  same  sensation  of  direction; 
every  point  evokes  the  mean  of  the  space-sensations 
of  the  neighboring  points;  every  part,  however 
small,  is  similar  to  every  other  part,  however  great. 
But,  though  it  has  influenced  the  definitions  of  many 
writers,4  the  geometer  can  accomplish  little  with 
this  physiological  characterization.  The  visual  im- 
age must  be  enriched  by  physical  experience  con- 
cerning corporeal  objects,  to  be  geometrically  avail- 
able. Let  a  string  be  fastened  by  one  extremity  at  A, 
and  let  its  other  extremity  be  passed  through  a  ring 
fastened  at  B.  If  we  pull  on  the  extremity  at  B, 
we  shall  see  parts  of  the  string  which  before  lay 
between  A  and  B  pass  out  at  B,  while  at  the  same 


lLoc.  cit.,  pp.  206-207. 

'Lot  cit. 

*  Anschauung. 

4  Euclid,  Elements,  I.,  Definition  3. 


62  SPACE    AND    GEOMETRY 

time  the  string  will  approach  the  form  of  a  straight 
line.  A  smaller  number  of  like  parts  of  the  string, 
identical  bodies,  suffices  to  compose  the  straight  line 
joining  A  and  B  than  to  compose  a  curved  line. 

It  is  erroneous  to  assert  that  the  straight  line  is 
recognized  as  the  shortest  line  by  mere  visualization. 
It  is  quite  true  we  can,  so  far  as  quality  is  concerned, 
reproduce  in  imagination  with  perfect  accuracy  and 
reliability,  the  simultaneous  change  of  form  and 
length  which  the  string  undergoes.  But  this  is 
nothing  more  than  a  reviviscence  of  a  prior  experi- 
ence with  bodies, — an  experiment  in  thought.  The 
mere  passive  contemplation  of  space  would  never 
lead  to  such  a  result.  Measurement  is  experience  in- 
volving a  physical  reaction,  an  experiment  of  super- 
position. Visualized  or  imagined  lines  having  dif- 
ferent directions  and  lengths  cannot  be  applied  to 
one  another  forthwith.  The  possibility  of  such 
a  procedure  must  be  actually  experienced  with  ma- 
terial objects  accounted  as  unalterable.  It  is  erron- 
eous to  attribute  to  animals  an  instinctive  knowledge 
of  the  straight  line  as  the  shortest  distance  between 
two  points.  If  a  stimulus  excites  an  animal's  at- 
tention, and  if  the  animal  has  so  turned  that  its 
plane  of  symmetry  passes  through  the  stimulating 
object,  then  the  straight  line  is  the  path  of  motion 
uniquely  determined  by  the  stimulus.  This  is  dis- 
tinctly shown  in  Loeb's  investigations  on  the  trop- 
isms  of  animals. 

Further,  visualization  alone  does  not  prove  that 
any  two  sides  of  a  triangle  are  together  greater 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY    63 

than  the  third  side.  It  is  true  that  if  the  two  sides 
be  laid  upon  the  base  by  rotation  round  the  vertices 
of  the  basal  angles,  it  will  be  seen  by  an  act  of  imagi- 
nation alone  that  the  two  sides  with  their  free  ends 
moving  in  arcs  of  circles  will  ultimately  overlap, 
thus  more  than  filling  up  the  base.  But  we  should 
not  have  attained  to  this  representation  had  not  the 
procedure  been  actually  witnessed  in  connection 
with  corporeal  objects.  Euclid1  deduces  this  truth 
circuitously  and  artificially  from  the  fact  that  the 
greater  side  of  every  triangle  is  opposite  to  the 
greater  angle.  But  the  source  of  our  knowledge 
here  also  is  experience, — experience  of  the  motion 
of  the  side  of  a  physical  triangle;  this  source  has, 
however,  been  laboriously  concealed  by  the  form  of 
the  deduction, — and  this  not  to  the  enhancement  of 
perspicuity  or  brevity. 

But  the  properties  of  the  straight  line  are  not  ex- 
hausted with  the  preceding  empirical  truths.  If  a 
wire  of  any  arbitrary  shape  be  laid  on  a  board  in 
contact  with  two  upright  nails,  and  slid  along  so  as 
to  be  always  in  contact  with  the  nails,  the  form  and 
position  of  the  parts  of  the  wire  between  the  nails 
will  be  constantly  changing.  The  straighter  the 
wire  is,  the  slighter  the  alteration  will  be.  A  straight 
wire  submitted  to  the  same  operation  slides  in  itself. 
Rotated  round  two  of  its  own  fixed  points,  a  crooked 
wire  will  keep  constantly  changing  its  position,  but 
a  straight  wire  will  maintain  its  position,  it  will  ro- 


1  Euclid,  Elements,  Book  I.,  Prop.  20. 


64  SPACE    AND    GEOMETRY 

tate  within  itself.1  When  we  define,  now,  a  straight 
line  as  the  line  which  is  completely  determined  by 
two  of  its  points,  there  is  nothing  in  this  concept 
except  the  idealisation  of  the  empirical  notion  de- 
rived from  the  physical  experience  mentioned, — a 
notion  by  no  means  directly  furnished  by  the  physi- 
ological act  of  visualization. 

The  plane,  like  the  straight  line,  is  physiologically 
characterized  by  its  simplicity.  It  appears  the  same 
at  all  parts.2  Every  point  evokes  the  mean  of  the 
space-sensations  of  the  neighboring  points.  Every 
part,  however  small,  is  like  every  other  part,  how- 
ever great.  But  experiences  gained  in  connection 
with  physical  objects  are  also  required,  if  these  prop- 
erties are  to  be  put  to  geometrical  account.  The 
plane,  like  the  straight  line,  is  physiologically  sym- 
metrical with  respect  to  itself,  if  it  coincides  with 
the  median  plane  of  the  body  or  stands  at  right 
angles  to  the  same.  But  to  discover  that  symmetry 
is  a  permanent  geometrical  property  of  the  plane 
and  the  straight  line,  both  concepts  must  be  given 
as  movable,  unalterable  physical  objects.  The 
connection  of  physiological  symmetry  with  metrical 


*In  a  letter  to  Vitale  Giordano  (Leibnizens  mathematische 
Schriften,  herausgegeben  v.  Gerhardt,  erste  Abtheilung,  Bd.  1., 
8.  195-196),  Leibnitz  makes  use  of  the  above-mentioned  prop- 
erty of  a  straight  line  for  its  definition.  The  straight  lin« 
shares  the  property  of  displaceability  in  itself  with  the  circle 
and  the  circular  cylindrical  spiral.  But  the  property  of  rotata- 
bility  within  itself  and  that  of  being  determined  by  two  points, 
are  exclusively  its  own. 

•  Compare  Euclid,  Elements  I.,  Definition  7. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY    6$ 


properties  also  is  in  need  of  special  metrical  demon- 
stration. 

Physically  a  plane  is  constructed  by  rubbing  three 
bodies  together  until  three  surfaces,  A,  B,  C,  are  ob- 
tained, each  of  which  exactly  fits  the  others, — a  re- 
sult which  can  be  accomplished,  as  Fig.  9  shows, 
with  neither  convex  nor  concave  surfaces,  but  with 
plane  surfaces  only.  The  convexities  and  concavi- 
ties are,  in  fact,  removed  by  the  rubbing.  Similarly, 
a  truer  straight  line  can  be  obtained  with  the  aid  of 
an  imperfect  ruler,  by  first  placing  it  with  its  ends 
against  the  points  A,  B,  then  turning  it  through  an 
angle  of  180°  out  of  its  plane  and  again  placing  it 
against  A,  B,  afterwards  taking  the  mean  between 
the  two  lines  so  obtained  as  a  more  perfect  straight 
line,  and  repeating  the  operation  with  the  line  last 


Fig.  9. 


obtained.  Having  by  rubbing,  produced  a  plane, 
that  is  to  say,  a  surface  having  the  same  form  at 
all  points  and  on  both  sides,  experience  furnishes  ad- 
ditional results.  Placing  two  such  planes  one  on 


66  SPACE    AND    GEOMETRY 

the  other,  it  will  be  learned  that  the  plane  is  dis- 
placeable  into  itself,  and  rotatable  within  itself,  just 
as  a  straight  line  is.  A  thread  stretched  between 
any  two  points  in  the  plane  falls  entirely  within  the 
plane.  A  piece  of  cloth  drawn  tight  across  any 
bounded  portion  of  a  plane  coincides  with  it.  Hence 
the  plane  represents  the  minimum  of  surface  within 
its  boundaries.  If  the  plane  be  laid  on  two  sharp 
points,  it  can  still  be  rotated  around  the  straight 
line  joining  the  points,  but  any  third  point  outside 
of  this  straight  line  fixes  the  plane,  that  is,  deter- 
mines it  completely. 

In  the  letter  to  Vitale  Giordano,  above  referred 
to,  Leibnitz  makes  the  frankest  use  of  this  experi- 
ence with  corporeal  objects,  when  he  defines  a  plane 
as  a  surface  which  divides  an  unbounded  solid  into 
two  congruent  parts,  and  a  straight  line  as  a  line 
which  divides  an  unbounded  plane  into  two  con- 
gruent parts.1 

If  attention  be  directed  to  the  symmetry  of  the 
plane  with  respect  to  itself,  and  two  points  be  as- 
sumed, one  on  each  side  of  it,  each  symmetrical  to 
the  other,  it  will  be  found  that  every  point  in  the 
plane  is  equidistant  from  these  two  points,  and  Leib- 


»The  passage  reads  literally:  "Et  difficulter  absolvi  poterit 
demonstratio,  nisi  quis  assumat  notionem  rectse,  qualis  est  qua 
ego  uti  soleo,  quod  corpore  aliquo  duobus  punctis  immotis 
revoluto  locus  omnium  punctorum  quiescentium  sit  recta,  vel 
saltern  quod  recta  sit  linea  secans  planum  interminatum  in  duas 
partes  congruas;  et  planum  sit  superficies  secans  solidum  inter- 
minatum in  duas  partes  congruas. ' '  For  similar  definitions,  see, 
for  example,  Halsted's  Elements  of  Geometry,  6th  edition. 
New  York,  1895,  p.  9.— T.  J.  McC. 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     67 

nitz's  definition  of  the  plane  is  reached.1  The  uni- 
formity and  symmetry  of  the  straight  line  and  the 
plane  are  consequences  of  their  being  absolute  min- 
ima of  length  and  area  respectively.  For  the 
boundaries  given  the  minimum  must  exist,  no  other 
collateral  condition  being  involved.  The  minimum 
is  unique,  single  in  its  kind;  hence  the  symmetry 
with  respect  to  the  bounding  points.  Owing  to  the 
absoluteness  of  the  minimum,  every  portion,  how- 
ever small,  again  exhibits  the  same  minimal  prop- 
erty; hence  the  uniformity. 

EMPIRICAL  ORIGIN  OF  GEOMETRY. 
Empirical  truths  organically  connected  may  make 
their  appearance  independently  of  one  another,  and 
doubtless  were  so  discovered  long  before  the  fact  of 
their  connection  was  known.  But  this  does  not  pre- 
clude their  being  afterwards  recognized  as  involved 
in,  and  determined  by,  one  another,  as  being  de- 
ducible  from  one  another.  For  example,  supposing 
we  are  acquainted  with  the  symmetry  and  uniform- 
ity of  the  straight  line  and  the  plane,  we  easily 
deduce  that  the  intersection  of  two  planes  is  a 
straight  line,  that  any  two  points  of  the  plane  can 
be  joined  by  a  straight  line  lying  wholly  within  the 
plane,  etc.  The  fact  that  only  a  minimum  of  incon- 
spicuous and  unobtrusive  experiences  is  requisite 
for  such  deductions  should  not  lure  us  into  the  er- 
ror of  regarding  this  minimum  as  wholly  super- 

1  Leibnitz,  in  re  "  geometrical  characteristic, ' '  letter  to  Huy- 
gena,  Sept.  8.  1679  (Gerhardt,  loc.  tit.,  erste  Abth.,  Bd.  II., 
8.  23). 


68  SPACE    AND    GEOMETRY 

fluous,  and  of  believing  that  visualization  and  rea- 
soning are  alone  sufficient  for  the  construction  of 
geometry. 

Like  the  concrete  visual  images  of  the  straight 
line  and  the  plane,  so  also  our  visualizations  of  the 
circle,  the  sphere,  the  cylinder,  etc.,  are  enriched 
by  metrical  experiences,  and  in  this  manner  first 
rendered  amenable  to  fruitful  geometrical  treatment. 
The  same  economic  impulse  that  prompts  our  chil- 
dren to  retain  only  the  typical  features  in  their  con- 
cepts and  drawings,  leads  us  also  to  the  schematisa- 
tion  and  conceptual  idealisation  of  the  images  de- 
rived from  our  experience.  Although  we  never 
come  across  in  nature  a  perfect  straight  line  or  an 
exact  circle,  in  our  thinking  we  nevertheless  de- 
signedly abstract  from  the  deviations  which  thus 
occur.  Geometry,  therefore,  is  concerned  with  ideal 
objects  produced  by  the  schematization  of  experi- 
ential objects.  I  have  remarked  elsewhere  that  it 
is  wrong  in  elementary  geometrical  instruction  to 
cultivate  predominantly  the  logical  side  of  the  sub- 
ject, and  to  neglect  to  throw  open  to  young  students 
the  wells  of  knowledge  contained  in  experience.  It 
is  gratifying  to  note  that  the  Americans  who  are 
less  dominated  than  we  by  tradition,  have  recently 
broken  with  this  system  and  are  introducing  a  sort 
of  experimental  geometry  as  introductory  to  sys- 
tematic geometric  instruction.* 

*See  the  essays  and  books  of  Harms,  Campbell,  Speer,  Myers, 
Hall  and  many  others  noticed  in  the  reviews  of  School  Science 
and  Mathematics  (Chicago)  during  the  last  few  years. — T.  J. 
McC. 


PSYCHOLOGY  AND  DEVELOPMENT  OF  GEOMETRY  69 

TECHNICAL  AND  SCIENTIFIC  DEVELOPMENT  OF 
GEOMETRY. 

No  sharp  line  can  be  drawn  between  the  instinc- 
tive, the  technical,  and  the  scientific  acquisition  of 
geometric  notions.  Generally  speaking,  we  may  say, 
perhaps,  that  with  division  of  labor  in  the  indus- 
trial and  economic  fields,  with  increasing  employ- 
ment with  very  definite  objects,  the  instinctive  acqui- 
sition of  knowledge  falls  into  the  background,  and 
the  technical  begins.  Finally,  when  measurement 
becomes  an  aim  and  business  in  itself,  the  connec- 
tion obtaining  between  the  various  operations  of 
measuring  acquires  a  powerful  economic  interest, 
and  we  reach  the  period  of  the  scientific  develop- 
ment of  geometry,  to  which  we  now  proceed. 

The  insight  that  the  measures  of  geometry  de- 
pend on  one  another,  was  reached  in  divers  ways. 
After  surfaces  came  to  be  measured  by  surfaces, 
further  progress  was  almost  inevitable.  In  a  paral- 
lelogrammatic  field  permitting  a  division  into  equal 
partial  parallelogrammatic  fields  so  that  n  rows  of 
partial  fields  each  containing  m  fields  lay  alongside 
one  another,  the  counting  of  these  fields  was  un- 
necessary. By  multiplying  together  the  numbers 
measuring  the  sides,  the  area  of  the  field  was  found 
to  be  equal  to  win  such  fields,  and  the  area  of  each 
of  the  two  triangles  formed  by  drawing  the  diago- 
nal was  readily  discovered  to  be  equal  to  ^  such 
fields.  This  was  the  first  and  simplest  application  of 


7O  SPACE    AND    GEOMETRY 

arithmetic  to  geometry.  Coincidently,  the  depend- 
ence of  measures  of  area  on  other  measures,  linear 
and  angular,  was  discovered.  The  area  of  a  rec- 
tangle was  found  to  be  larger  than  that  of  an  ob- 
lique parallelogram  having  sides  of  the  same  length ; 
the  area,  consequently,  depended  not  only  on  the 
length  of  the  sides,  but  also  on  the  angles.  On  the 
other  hand,  a  rectangle  constructed  of  strips  of 
wood  running  parallel  to  the  base,  can,  as  is  easily 
seen,  be  converted  by  displacement  into  any  paral- 
lelogram of  the  same  height  and  base  without  alter- 
ing its  area.  Quadrilaterals  having  their  sides  given 
are  still  undetermined  in  their  angles,  as  every  car- 
penter knows.  He  adds  diagonals,  and  converts 
liis  quadrilateral  into  triangles,  which,  the  sides  be- 
ing given,  are  rigid,  that  is  to  say,  are  unalterable 
as  to  their  angles  also. 

With  the  perception  that  measures  were  depend- 
ent on  one  another,  the  real  problem  of  geometry 
was  introduced.  Steiner  has  aptly  and  justly  en- 
titled his  principal  work  "Systematic  Development 
of  the  Dependence  of  Geometrical  Figures  on  One 
Another."1  In  Snell's  original  but  unappreciated 
treatise  on  Elementary  Geometry,  the  problem  in 
question  is  made  obvious  even  to  the  beginner.2 

A  plane  physical  triangle  is  constructed  of  wires. 
If  one  of  the  sides  be  rotated  around  a  vertex,  so  as 
to  increase  the  interior  angle  at  that  point,  the  side 


1 J.  Steiner,  Systematise!*  EntwicTclung  der  Abhiingiglceit  der 
geometrischen  Gestalten  von  einander. 
1  Snell,  Lehrbuch  der  Geometric,  Leipsic,  1869. 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     /I 

moved  will  be  seen  to  change  its  position  and  the 
side  opposite  to  grow  larger  with  the  angle.  New 
pieces  of  wire  besides  those  before  present  will  be 
required  to  complete  the  last-mentioned  side.  This 
and  other  similar  experiments  can  be  repeated  in 
thought,  but  the  mental  experiment  is  never  any- 
thing more  than  a  copy  of  the  physical  experiment. 
The  mental  experiment  would  be  impossible  if  phys- 
ical experience  had  not  antecedently  led  us  to  a 
knowledge  of  spatially  unalterable  physical  bodies,1 
— to  the  concept  of  measure. 

THE  GEOMETRY  OF  THE  TRIANGLE. 

By  experiences  of  this  character,  we  are  conducted 
to  the  truth  that  of  the  six  metrical  magnitudes  dis- 
coverable in  a  triangle  (three  sides  and  three  angles) 
three,  including  at  least  one  side,  suffice  to  determine 
the  triangle.  If  one  angle  only  be  given  among  the 
parts  determining  the  triangle,  the  angle  in  question 
must  be  either  the  angle  included  by  the  given  sides, 
or  that  which  is  opposite  to  the  greater  side, — at 
least  if  the  determination  is  to  be  unique.  Having 
reached  the  perception  that  a  triangle  is  determined 
by  three  sides  and  that  its  form  is  independent  of  its 
position,  it  follows  that  in  an  equilateral  triangle  all 
three  angles  and  in  an  isosceles  triangle  the  two 
angles  opposite  the  equal  sides,  must  be  equal,  in 


*The  whole  construction  of  the  Euclidean  geometry  shows 
traces  of  this  foundation.  It  is  still  more  conspicuous  in  the 
"geometric  characteristic'*  of  Leibnitz  already  mentioned. 
We  shall  revert  to  this  topic  later. 


72  SPACE    AND    GEOMETRY 

whatever  manner  the  angles  and  sides  may  depend 
on  one  another.  This  is  logically  certain.  But  the 
empirical  foundation  on  which  it  rests  is  for  that 
reason  not  a  whit  more  superfluous  than  it  is  in  the 
analogous  cases  of  physics. 

The  mode  in  which  the  sides  and  angles  depend 
on  one  another  is,  naturally,  first  recognized  in  spe- 
cial instances.  In  computing  the  areas  of  rectangles 
and  of  the  triangles  formed  by  their  diagonals,  the 
fact  must  have  been  noticed  that  a  rectangle  having 
sides  3  and  4  units  in  length  gives  a  right-angled 
triangle  having  sides,  3,  4,  and  5  units  in  length. 
Rectangularity  was  thus  shown  to  be  connected  with 
a  definite,  rational  ratio  between  the  sides.  The 
knowledge  of  this  truth  was  employed  to  stake  off 
right  angles,  by  means  of  three  connected  ropes 
respectively  3,  4,  and  5  units  in  length.1  The  equa- 
tion 32  +  42  =  52,  the  analogue  of  which  was  proved 
to  be  valid  for  all  right-angled  triangles  having  sides 
of  lengths  a,  bf  c  (the  general  formula  being 
a2  +  b2  =  c2),  now  riveted  the  attention.  It  is  well 
known  how  profoundly  this  relation  enters  into  met- 
rical geometry,  and  how  all  indirect  measurements 
of  distance  may  be  traced  back  to  it.  We  shall  en- 
deavor to  disclose  the  foundation  of  this  relation. 

It  is  to  be  remarked  first  that  neither  the  Greek 
geometrical  nor  the  Hindu  arithmetical  deductions 
of  the  so-called  Pythagorean  Theorem  could  avoid 
the  consideration  of  areas.  One  essential  point  on 


1  Cantor,  Geschichte  der  Mathematik,  Leipsic,  1880.     I.,  pp. 
53,  56. 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     73 

which  all  the  deductions  rest  and  which  appears  more 
or  less  distinctly  in  different  forms  in  all  of  them,  is 
the  following.  If  a  triangle,  a,  b,  c  (Fig.  10)  be  slid 
along  a  short  distance  in  its  own  plane,  it  is  as- 


Fig.  10. 

sumed  that  the  space  which  it  leaves  behind  is  com- 
pensated for  by  the  new  space  on  which  it  enters. 
That  is  to  say,  the  area  swept  out  by  two  of  the  sides 
during  the  displacement  is  equal  to  the  area  swept 
out  by  the  third  side.  The  basis  of  this  conception 
is  the  assumption  of  the  conservation  of  the  area  of 
the  triangle.  If  we  consider  a  surface  as  a  body  of 
very  minute  but  unvarying  thickness  of  third  dimen- 
sion (which  for  that  reason  is  uninflueritial  in  the 
present  connection),  we  shall  again  have  the  con- 
servation of  the  volume  of  bodies  as  our  funda- 
mental assumption.  The  same  conception  may  be 
applied  to  the  translation  of  a  tetrahedron,  but  it 
does  not  lead  in  this  instance  to  new  points  of  view. 
Conservation  of  volume  is  a  property  which  rigid 
and  liquid  bodies  possess  in  common,  and  was  ideal- 
ized by  the  old  physics  as  impenetrability.  In  the 
case  of  rigid  bodies,  we  have  the  additional  at- 
tribute that  the  distances  between  all  the  parts  are 
preserved,  while  in  the  case  of  liquids,  the  proper- 


74 


SPACE    AND    GEOMETRY 


ties  of  rigid  bodies  exist  only  for  the  smallest  time 
and  space  elements. 

If  an  oblique-angled  triangle  having  the  sides 
a,  b,  and  c  be  displaced  in  the  direction  of  the  side 
bf  only  a  and  c  will,  by  the  principle  above  stated, 
describe  equivalent  parallelograms,  which  are  alike 
in  an  equal  pair  of  parallel  sides  on  the  same  paral- 
lels. If  a  make  with  b  a  right  angle,  and  the  tri- 
angle be  displaced  at  right  angles  to  c,  the  distance 
c,  the  side  c  will  describe  the  square  c2,  while  the 
two  other  sides  will  describe  parallelograms  the 
combined  areas  of  which  are  equal  to  the  area  of  the 


Fig.  11. 

square.  But  the  two  parallelograms  are,  by  the  ob- 
servation which  just  precedes,  equivalent  respectively 
to  a2  and  b2, — and  with  this  the  Pythagorean  the- 
orem is  reached. 

The  same  result  may  also  be  attained  ( Fig.  1 1 ) 
by  first  sliding  the  triangle  a  distance  a  at  right  an- 
gles to  a,  and  then  a  distance  b  at  right  angles  to 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY    75 

b,  where  a2  +  b2  will  be  equal  to  the  sum  of 
the  surfaces  swept  out  by  c,  which  is  obviously  c2. 
Taking  an  oblique-angled  triangle,  the  same  proced- 
ure just  as  easily  and  obviously  gives  the  more  gen- 
eral proposition,  c2  =  a2  +  b2  —  2afrcosy. 

The  dependence  of  the  third  side  of  the  triangle 
on  the  two  other  sides  is  accordingly  determined  by 
the  area  of  the  enclosed  triangle ;  or,  in  our  concep- 
tion, by  a  condition  involving  volume.  It  will  also 
be  directly  seen  that  the  equations  in  question  ex- 
press relations  of  area.  It  is  true  that  the  angle  in- 
cluded between  two  of  the  sides  may  also  be  re- 
garded as  determinative  of  the  third  side,  in  which 
case  the  equations  will  aparently  assume  an  en- 
tirely different  form. 

Let  us  look  a  little  more  closely  at  these  different 
measures.  If  the  extremities  of  two  straight  lines 
of  lengths  a  and  b  meet  in  a  point,  the  length  of  the 
line  c  joining  their  free  extremities  will  be  included 
between  definite  limits.  We  shall  have  c  <  a  +  b, 
and  c  ^>  a  —  b.  Visualization  alone  cannot  inform 
us  of  this  fact;  we  can  learn  it  only  from  experi- 
menting in  thought, — a  procedure  which  both  re- 
poses on  physical  experience  and  reproduces  it.  This 
will  be  seen  by  holding  a  fast,  for  example,  and 
turning  b,  first,  until  it  forms  the  prolongation  of  a, 
and,  secondly,  until  it  coincides  with  a.  A  straight 
line  is  primarily  a  unique  concrete  image  character- 
ized by  physiological  properties, — an  image  which 
we  have  obtained  from  a  physical  body  of  a  definite 


76  SPACE    AND    GEOMETRY 

specific  character,  which  in  the  form  of  a  string  or 
wire  of  indefinitely  small  but  constant  thickness  in- 
terposes a  minimum  of  volume  between  the  posi- 
tions of  its  extremities, — which  can  be  accomplished 
only  in  one  uniquely-determined  manner.  If  sev- 
eral straight  lines  pass  through  a  point,  we  distin- 
guish between  them  physiologically  by  their  direc- 
tions. But  in  abstract  space  obtained  by  metrical 
experiences  with  physical  objects,  differences  of  di- 
rection do  not  exist.  A  straight  line  passing 
through  a  point  can  be  completely  determined  in  ab- 
stract space  only  by  assigning  a  second  physical 
point  on  it.  To  define  a  straight  line  as  a  line  which 
is  constant  in  direction,  or  an  angle  as  a  difference 
between  directions,  or  parallel  straight  lines  as 
straight  lines  having  the  same  direction,  is  to  define 
these  concepts  physiologically. 

THE  MEASUREMENT  OF  THE  ANGLE. 

Different  methods  are  at  our  disposal  when  we 
come  to  characterize  or  determine  geometrically  an- 
gles which  are  visually  given.  An  angle  is  deter- 
mined when  the  distance  is  assigned  between  any 
two  fixed  points  lying  each  on  a  separate  side  of 
the  angle  outside  the  point  of  intersection.  To  ren- 
der the  definition  uniform,  points  situated  at  the 
same  fixed  and  invariable  distance  from  the  vertex 
might  be  chosen.  The  inconvenience  that  then  equi- 
multiples of  a  given  angle  placed  alongside  one  an- 
other in  the  same  plane  with  their  vertices  coinci- 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     7/ 

dent,  would  not  be  measured  by  the  same  equimulti- 
ples of  the  distance  between  those  points,  is  the  rea- 
son that  this  method  of  determining  angles  was  not 
introduced  into  elementary  geometry.1  A  simpler 
measure,  a  simpler  characterization  of  an  angle,  is 
obtained  by  taking  the  aliquot  part  of  the  circumfer- 
ence or  the  area  of  a  circle  which  the  angle  inter- 
cepts when  laid  in  the  plane  of  the  circle  with  its 
vertex  at  the  center.  The  convention  here  involved 
is  more  convenient.2 

In  employing  an  arc  of  a  circle  to  determine  an 
angle,  we  are  again  merely  measuring  a  volume, — 
viz.,  the  volume  occupied  by  a  body  of  simple  defi- 
nite form  introduced  between  two  points  on  the 
arms  of  the  angle  equidistant  from  the  vertex.  But 
a  circle  can  be  characterized  by  simple  rectilinear 
distances.  It  is  a  matter  of  perspicuity,  of  immedi- 
acy, and  of  the  facility  and  convenience  resulting 
therefrom,  that  two  measures,  viz.,  the  rectilinear 
measure  of  length  and  the  angular  measure,  are 
principally  employed  as  fundamental  measures,  and 
that  the  others  are  derived  from  them.  It  is  in  no 
sense  necessary.  For  example  (Fig.  12),  it  is  possi- 
ble without  a  special  angular  measure  to  determine 
the  straight  line  that  cuts  another  straight  line  at 
right  angles  by  making  all  its  points  equidistant 
from  two  points  in  the  first  straight  line  lying  at 
equal  distances  from  the  point  of  intersection.  The 

1 A  closely  allied  principle  of  measurement  is,  however,  ap- 
plied in  trigonometry. 

1  So  also  the  superficial  portion  of  a  sphere  intercepted  by  the 
including  planes  is  used  as  the  measure  of  a  solid  angle. 


78  SPACE    AND    GEOMETRY 

bisector  of  an  angle  can  be  determined  in  a  quite 
similar  manner,  and  by  continued  bisection  an 
angular  unit  can  be  derived  of  any  smallness  we 
wish.  A  straight  line  parallel  to  another  straight 


Fig.  12. 

line  can  be  defined  as  one,  all  of  whose  points  can 
be  translated  by  congruent  curved  or  straight  paths 
into  points  of  the  first  straight  line.1 

LENGTH  AS  THE  FUNDAMENTAL  MEASURE. 
It  is  quite  possible  to  start  with  the  straight 
length  alone  as  our  fundamental  measure.  Let  a 
fixed  physical  point  a  be  given.  Another  point,  m, 
has  the  distance  ra  from  the  first  point.  Then  this 
last  point  can  still  lie  in  any  part  of  the  spherical 
surface  described  about  a  with  radius  ra.  If  we 
know  still  a  second  fixed  point  b,  from  which  m  is 
removed  by  the  distance  r6,  the  triangle  abm  will 


*If  this  form  had  been  adopted,  the  doubts  as  to  the  Eucli- 
dean theorem  of  parallels  would  probably  have  risen  much  later. 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     79 

be  rigid,  determined;  but  m  can  still  revolve  round 
in  the  circle  described  by  the  rotation  of  the  triangle 
around  the  axis  ab.  If  now  the  point  m  be  held  fast 
in  any  position,  then  also  the  whole  rigid  body  to 
which  the  three  points  in  question,  a,  b,  m,  belong 
will  be  fixed. 

A  point  m  is  spatially  determined,  accordingly, 
by  the  distances  ra,  r^,  rc  from  at  least  three  fixed 
points  in  space,  af  b,  c.  But  this  determination  is 
still  not  unique,  for  the  pyramid  with  the  edges 
fa,  r*9rc9  in  the  vertex  of  which  m  lies,  can  be  con- 
structed as  well  on  the  one  as  on  the  other  side  of 
the  plane  a,  b,  c.  If  we  were  to  fix  the  side,  say  by 
a  special  sign,  we  should  be  resorting  to  a  physiolog- 
ical determination,  for  geometrically  the  two  sides 
of  the  plane  are  not  different.  If  the  point  m  is  to 
be  uniquely  determined,  its  distance,  rd,  from  a 
fourth  point,  dt  lying  outside  the  plane  abcf  must  in 
addition  be  given.  Another  point,  m',  is  determined 
with  like  completeness  by  four  distances,  r*M  r'  6, 
r'e,  r' d.  Hence,  the  distance  of  m  from  m'  is  also 
given  by  this  determination.  And  the  same  holds 
true  of  any  number  of  other  points  as  severally  de- 
termined by  four  distances.  Between  four  points 

4(4 —  0 

• =  6  distances  are  conceivable,  and  pre- 
cisely this  number  must  be  given  to  determine  the 
form  of  the  point  complex.  For  4  +  2  =  n  points, 
6  +  4,2  or  4n  —  10  distances  are  needed  for  the  de- 
termination, while  a  still  larger  number,  viz., 


80  SPACE    AND    GEOMETRY 

*  (ft—1)  distances  exist,  so  that  the  excess  of  the 

distances  is  also  coincidently  determined.* 

If  we  start  from  three  points  and  prescribe  that 
the  distances  of  all  points  to  be  further  determined 
shall  hold  for  one  side  only  of  the  plane  determined 
by  the  three  points,  then  $n  —  6  distances  will  suf- 
fice to  determine  the  form,  magnitude,  and  position 
of  a  system  of  n  points  with  respect  to  the  three 
initial  points.  But  if  there  be  no  condition  as  to  the 
side  of  the  plane  to  be  taken, — a  condition  which 
involves  sensuous  and  physiological,  but  not  ab- 
stract metrical  characteristics, — the  system  of 
points,  instead  of  the  intended  form  and  position, 
may  assume  that  symmetrical  to  the  first,  or  be  com- 
bined of  the  points  of  both.  Symmetric  geometrical 
figures  are,  owing  to  our  symmetric  physiological 
organisation,  very  easily  taken  to  be  identical, 
whereas  metrically  and  physically  they  are  entirely 
different.  A  screw  with  its  spiral  winding  to  the 
right  and  one  with  its  spiral  winding  to  the  left,  two 
bodies  rotating  in  contrary  directions,  etc.,  appear 
very  much  alike  to  the  eye.  But  we  are  for  this  rea- 
son not  permitted  to  regard  them  as  geometrically 
or  physically  equivalent.  Attention  to  this  fact 
would  avert  many  paradoxical  questions.  Think 
only  of  the  trouble  that  such  problems  gave  Kant! 


*For  an  interesting  attempt  to  found  both  the  Euclidean 
and  non-Euclidean  geometries  on  the  pure  notion  of  distance, 
see  De  Tilly,  "Essai  sur  les  principes  fondamentaux  de  la 
g6ometrie  et  de  la  mgcanique"  (Memoires  de  la  Sotitte  de 
Bordeaux,  1880). 


PSYCHOLOGY   AND   DEVELOPMENT   OF    GEOMETRY    8l 

Sensuous  physiological  attributes  are  determined  by 
relationship  to  our  body,  to  a  corporeal  system  of 
specific  constitution;  while  metrical  attributes  are 
determined  by  relations  to  the  world  of  physical 
bodies  at  large.  The  latter  can  be  ascertained  only 
by  experiments  of  coincidence, — by  measurements. 

VOLUME  THE  BASIS  OF  MEASUREMENT. 

As  we  see,  every  geometrical  measurement  is  at 
bottom  reducible  to  measurements  of  volumes,  to 
the  enumeration  of  bodies.  Measurements  of 
lengths,  like  measurements  of  areas,  repose  on  the 
comparison  of  the  volumes  of  very  thin  strings, 
sticks,  and  leaves  of  constant  thickness.  This  is  not 
at  variance  with  the  fact  that  measures  of  area  may 
be  arithmetically  derived  from  measures  of  length, 
or  solid  measures  from  measures  of  length  alone,  or 
from  these  in  combination  with  measures  of  area. 
This  is  merely  proof  that  different  measures  of  vol- 
ume are  dependent  on  one  another.  To  ascertain 
the  forms  of  this  interdependence  is  the  fundamen- 
tal object  of  geometry,  as  it  is  the  province  of  arith- 
metic to  ascertain  the  manner  in  which  the  various 
numerical  operations,  or  ordinative  activities  of  the 
mind,  are  connected  together. 

THE  VISUAL  SENSE  IN  GEOMETRY. 

It  is  extremely  probable  that  the  experiences  of 
the  visual  sense  were  the  cause  of  the  rapidity  with 
which  geometry  developed.  But  our  great  famil- 


82  SPACE    AND    GEOMETRY 

iarity  with  the  properties  of  rays  of  light  gained 
from  the  present  advanced  state  of  optical  tech- 
nique, should  not  mislead  us  into  regarding  our 
experimental  knowledge  of  rays  of  light  as  the 
principal  foundation  of  geometry.  Rays  of  light  in 
dust  or  smoke-laden  air  furnish  admirable  visualiza- 
tions of  straight  lines.  But  we  can  derive  the  met- 
rical properties  of  straight  lines  from  rays  of  light 
just  as  little  as  we  can  derive  them  from  imaged 
straight  lines.  For  this  purpose  experiences  with 
physical  objects  are  absolutely  necessary.  The  rope- 
stretching  of  the  practical  geometers  is  certainly 
older  than  the  use  of  the  theodolite.  But  once 
knowing  the  physical  straight  line,  the  ray  of  light 
furnishes  a  very  distinct  and  handy  means  of  reach- 
ing new  points  of  view.  A  blind  man  could  scarcely 
have  invented  modern  synthetic  geometry.  But  the 
oldest  and  the  most  powerful  of  the  experiences  ly- 
ing at  the  basis  of  geometry  are  just  as  accessible  to 
the  blind  man,  through  his  sense  of  touch,  as  they 
are  to  the  person  who  can  see.  Both  are  acquainted 
with  the  spatial  permanency  of  bodies  despite  their 
mobility;  both  acquire  a  conception  of  volume  by 
taking  hold  of  objects.  The  creator  of  primitive 
geometry  disregards,  first  instinctively  and  then 
intentionally  and  consciously,  those  physical  proper- 
ties that  are  unessential  to  his  operations  and  that 
for  the  moment  do  not  concern  him.  In  this  man- 
ner, and  by  gradual  growth,  the  idealized  concepts 
of  geometry  arise  on  the  basis  of  experience. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     83 

VARIOUS  SOURCES  OF  OUR  GEOMETRIC  KNOWL- 
EDGE. 

Our  geometrical  knowledge  is  thus  derived  from 
various  sources.  We  are  physiologically  acquainted, 
from  direct  visual  and  tactual  contact,  with  many 
and  various  spatial  forms.  With  these  are  asso- 
ciated physical  (metrical)  experiences  (involving 
comparison  of  the  space-sensations  evoked  by  dif- 
ferent bodies  under  the  same  circumstances),  which 
experiences  are  in  their  turn  also  but  the  expres- 
sions of  other  relations  obtaining  between  sensa- 
tions. These  diverse  orders  of  experience  are  so 
intimately  interwoven  with  one  another  that  they 
can  be  separated  only  by  the  most  thoroughgoing 
scrutiny  and  analysis.  Hence  originate  the  widely 
divergent  views  concerning  geometry.  Here  it  is 
based  on  pure  visualization  (Anschauung) ,  there  on 
physical  experience,  according  as  the  one  or  the 
other  factor  is  overrated  or  disregarded.  But  both 
factors  entered  into  the  development  of  geometry 
and  are  still  active  in  it  to-day;  for,  as  we  have 
seen,  geometry  by  no  means  exclusively  employs 
purely  metrical  concepts. 

If  we  were  to  ask  an  unbiased,  candid  person  un- 
der what  form  he  pictured  space,  referred,  for  ex- 
ample, to  the  Cartesian  system  of  co-ordinates,  he 
would  doubtless  say :  I  have  the  image  of  a  system 
of  rigid  (form-fixed),  transparent,  penetrable,  con- 
tiguous cubes,  having  their  bounding  surfaces 
marked  only  by  nebulous  visual  and  tactual  per- 


84  SPACE    AND    GEOMETRY 

cepts, — a  species  of  phantom  cubes.  Over  and 
through  these  phantom  constructions  the  real  bod- 
ies or  their  phantom  counterparts  move,  conserving 
their  spatial  permanency  (as  above  defined), 
whether  we  are  concerned  with  practical  or  theoret- 
ical geometry,  or  phoronomy.  Gauss's  famous  in- 
vestigation of  curved  surfaces,  for  instance,  is  really 
concerned  with  the  application  of  infinitely  thin 
laminate  and  hence  flexible  bodies  to  one  another. 
That  diverse  orders  of  experience  have  co-op- 
erated in  the  formation  of  the  fundamental  concep- 
tions under  consideration,  cannot  be  gainsaid. 

THE  FUNDAMENTAL  FACTS  AND  CONCEPTS. 

Yet,  varied  as  the  special  experiences  are  from 
which  geometry  has  sprung,  they  may  be  reduced  to 
a  minimum  of  facts:  Movable  bodies  exist  having 
definite  spatial  permanency, — viz.,  rigid  bodies  ex- 
ist. But  the  movability  is  characterized  as  follows : 
we  draw  from  a  point  three  lines  not  all  in  the  same 
plane  but  otherwise  undetermined.  By  three  move- 
ments along  these  straight  lines  any  point  can  be 
reached  from  any  other.  Hence,  three  measure- 
ments or  dimensions,  physiologically  and  metrically 
characterized  as  the  simplest,  are  sufficient  for  all 
spatial  determinations.  These  are  the  fundamental 
facts.1 

The  physical  metrical  experiences,  like  all  experi- 


1  The  historical  development  of  this  conception  will  be  con- 
sidered in  another  place. 


PSYCHOLOGY   AND    DEVELOPMENT    OF    GEOMETRY     85 

ences  forming  the  basis  of  experimental  sciences, 
are  conceptualized, — idealized.  The  need  of  repre- 
senting the  facts  by  simple  perspicuous  concepts 
under  easy  logical  control,  is  the  reason  for  this. 
Absolutely  rigid,  spatially  invariable  bodies,  per- 
fect straight  lines  and  planes,  no  more  exist  than  a 
perfect  gas  or  a  perfect  liquid.  Nevertheless,  de- 
ferring the  consideration  of  the  deviations,  we  pre- 
fer to  work,  and  we  also  work  more  readily,  with 
these  concepts  than  with  others  that  conform  more 
closely  to  the  actual  properties  of  the  objects.  The- 
oretical geometry  does  not  even  need  to  consider 
these  deviations,  inasmuch  as  it  assumes  objects  that 
fulfil  the  requirements  of  the  theory  absolutely,  just 
as  theoretical  physics  does.  But  in  practical  geom- 
etry, where  we  are  concerned  with  actual  objects, 
we  are  obliged,  as  in  practical  physics,  to  consider 
the  deviations  from  the  theoretical  assumptions. 
But  geometry  has  still  the  advantage  that  every 
deviation  of  its  objects  from  the  assumptions  of  the 
theory  which  may  be  detected  can  be  removed; 
whereas  physics  for  obvious  reasons  cannot  con- 
struct more  perfect  gases  than  actually  exist  in 
nature.  For,  in  the  latter  case,  we  are  concerned 
not  with  a  single  arbitrarily  constructible  spatial 
property  alone,  but  with  a  relation  (occurring  in  na- 
ture and  independent  of  our  will)  between  pressure, 
volume,  and  temperature. 

The  choice  of  the  concepts  is  suggested  by  the 
facts ;  yet,  seeing  that  this  choice  is  the  outcome  of 
our  voluntary  reproduction  of  the  facts  in  thought, 


86  SPACE    AND    GEOMETRY 

some  free  scope  is  left  in  the  matter.  The  impor- 
tance of  the  concepts  is  estimated  by  their  range 
of  application.  This  is  why  the  concepts  of  the 
straight  line  and  the  plane  are  placed  in  the  fore- 
ground, for  every  geometrical  object  can  be  split  up 
with  sufficient  approximateness  into  elements 
bounded  by  planes  and  straight  lines.  The  par- 
ticular properties  of  the  straight  line,  plane,  etc., 
which  we  decide  to  emphasize,  are  matters  of  our 
own  free  choice,  and  this  truth  has  found  expres- 
sion in  the  various  definitions  that  have  been  given 
of  the  same  concept.1 

EXPERIMENTING  IN  THOUGHT. 

The  fundamental  truths  of  geometry  have  thus, 
unquestionably,  been  derived  from  physical  experi- 
ence, if  only  for  the  reason  that  our  visualizations 
and  sensations  of  space  are  absolutely  inaccessible 
to  measurement  and  cannot  possibly  be  made  the 
subject  of  metrical  experience.  But  it  is  no  less  in- 
dubitable that  when  the  relations  connecting  our 
visualizations  of  space  with  the  simplest  metrical 
experiences  have  been  made  familiar,  then  geomet- 
rical facts  can  be  reproduced  with  great  facility  and 
certainty  in  the  imagination  alone, — that  is  by  purely 
mental  experiment.  The  very  fact  that  a  continuous 
change  in  our  space-sensation  corresponds  to  a  con- 
tinuous metrical  change  in  physical  bodies,  enables 


1  Compare,  for  example,  the  definitions  of  the  straight  line 
given  by  Euclid  and  by  Archimedes. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY        / 

us  to  ascertain  by  imagination  alone  the  particular 
metrical  elements  that  depend  on  one  another.  Now, 
if  such  metrical  elements  are  observed  to  enter  dif- 
ferent constructions  having  different  positions  in  pre- 
cisely the  same  manner,  then  the  metrical  results 
will  be  regarded  as  equal.  The  case  of  the  isos- 
celes and  equilateral  triangles,  above  mentioned, 
may  serve  as  an  example.  The  geometric  mental 
experiment  has  advantage  over  the  physical,  only  in 
the  respect  that  it  can  be  performed  with  far  sim- 
pler experiences  and  with  such  as  have  been  more 
easily  and  almost  unconsciously  acquired. 

Our  sensuous  imagings  and  visualizations  of 
space  are  qualitative,  not  quantitative  nor  metrical. 
We  derive  from  them  coincidences  and  differences 
of  extension,  but  never  real  magnitudes.  Conceive, 
for  example,  Fig.  13,  a  coin  rolling  clockwise  down 
and  around  the  rim  of  another  fixed  coin  of  the 
same  size,  without  sliding.  Be  our  imagination  as 
vivid  as  it  will,  it  is  impossible  by  a  pure  feat  of  re- 
productive imagery  alone,  to  determine  here  the 
angle  described  in  a  full  revolution.  But  if  it  be 
considered  that  at  the  beginning  of  the  motion  the 
radii  a,  a'  lie  in  one  straight  line,  but  that  after  a 
quarter  revolution  the  radii  b,  b'  lie  in  a  straight 
line,  it  will  be  seen  at  once  that  the  radius  a'  now 
points  vertically  upwards  and  has  consequently  per- 
formed half  a  revolution.  The  measure  of  the  rev- 
olution is  obtained  from  metrical  concepts,  which 
fix  idealized  experiences  on  definite  physical  ob- 
jects, but  the  direction  of  the  revolution  is  retained 


88  SPACE    AND    GEOMETRY 

in  the  sensuous  imagination.  The  metrical  con- 
cepts simply  determine  that  in  equal  circles  equal 
angles  are  subtended  by  equal  arcs,  that  the  radii 
to  the  point  of  contact  lie  in  a  straight  line,  etc. 


Fig.  13. 

If  I  picture  to  myself  a  triangle  with  one  of  its 
angles  increasing,  I  shall  also  see  the  side  opposite 
the  angle  increasing.  The  impression  thus  arises 
that  the  interdependence  in  question  follows  a  priori 
from  a  feat  of  imagination  alone.  But  the  imagina- 
tion has  here  merely  reproduced  a  fact  of  experience. 
Measure  of  angle  and  measure  of  side  are  two  phys- 
ical concepts  applicable  to  the  same  fact, — concepts 
that  have  grown  so  familiar  to  us  that  they  have 
come  to  be  regarded  as  merely  two  different  at- 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     89 

tributes  of  the  same  imaged  group  of  facts,  and 
hence  appear  as  linked  together  of  sheer  necessity. 
Yet  we  should  never  have  acquired  these  concepts 
without  physical  experience. 

The  combined  action  of  the  sensuous  imagina- 
tion with  idealized  concepts  derived  from  experi- 
ence is  apparent  in  every  geometrical  deduction. 
Let  us  consider,  for  example,  the  simple  theorem 
that  the  perpendicular  bisectors  of  the  sides  of  a  tri- 
angle ABC  meet  in  a  common  point.  Experiment 
and  imagination  both  doubtless  led  to  the  theorem. 
But  the  more  carefully  the  construction  is  executed, 
the  more  one  becomes  convinced  that  the  third  per- 
pendicular does  not  pass  exactly  through  the  point 
of  intersection  of  the  first  two,  and  that  in  any  ac- 
tual construction,  therefore,  three  points  of  intersec- 
tion will  be  found  closely  adjacent  to  one  another. 
For  in  reality  neither  perfect  straight  lines  nor  per- 
fect perpendiculars  can  be  drawn;  nor  can  the  lat- 
ter be  erected  exactly  at  the  midpoints;  and  so  on. 
Only  on  the  assumption  of  these  ideal  conditions 
does  the  perpendicular  bisector  of  AB  contain  all 
points  equally  distant  from  A  and  B,  and  the  perpen- 
dicular bisector  of  BC  all  points  equidistant  from 
B  and  C.  From  which  it  follows  that  the  point  of 
intersection  of  the  two  is  equidistant  from  A,  B,  and 
C,  and  by  reason  of  its  equidistance  from  A  and  C 
is  also  a  point  of  the  third  perpendicular  bisector, 
of  AC.  The  theorem  asserts  therefore  that  the  more 
accurately  the  assumptions  are  fulfilled  the  more 
nearly  will  the  three  points  of  intersection  coincide. 


9O  SPACE    AND    GEOMETRY 

KANT'S  THEORY. 

The  importance  of  the  combined  action  of  the 
sensuous  imagination  [viz.,  of  the  Anschauung  or 
intuition  so  called]  and  of  concepts,  will  doubtless 
have  been  rendered  clear  by  these  examples.  Kant 
says:  "Thoughts  without  contents  are  empty,  in- 
tuitions without  concepts  are  blind."1  Possibly  we 
might  more  appropriately  say:  "Concepts  without 
intuitions  are  blind,  intuitions  without  concepts  are 
lame."  For  it  would  appear  to  be  not  so  absolutely 
correct  to  call  intuitions  [viz.,  sensuous  images] 
blind  and  concepts  empty.  When  Kant  further 
says  that  "there  is  in  every  branch  of  natural  knowl- 
edge only  so  much  science  as  there  is  mathematics 
contained  in  it,"2  one  might  possibly  also  assert  of 
all  sciences,  including  mathematics,  "that  they  are 
only  in  so  far  sciences  as  they  operate  with  con- 
cepts." For  our  logical  mastery  extends  only  to 
those  concepts  of  which  we  have  ourselves  deter- 
mined the  contents. 

THE  PRESENT  FORM  OF  GEOMETRY. 

The  two  facts  that  bodies  are  rigid  and  movable 
would  be  sufficient  for  an  understanding  of  any 
geometrical  fact,  no  matter  how  complicated, — suffi- 
cient, that  is  to  say,  to  derive  it  from  the  two  facts 


*Kritik  tier  reinen   Vernunft,   1787,   p.   75.     Max   Mtiller'a 
translation,  2nd  ed.,  1896,  p.  41. 

a  Metaphysische  Anfangsgriinde  der  Naturwissenschaft.    Vor- 
wort. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     9! 

mentioned.  But  geometry  is  obliged,  both  in  its 
own  interests  and  in  its  role  as  an  auxiliary  science, 
as  well  as  in  the  pursuit  of  practical  ends,  to  answer 
questions  that  recur  repeatedly  in  the  same  form. 
Now  it  would  be  uneconomical,  in  such  a  contin- 
gency, to  begin  each  time  with  the  most  elementary 
facts  and  to  go  to  the  bottom  of  each  new  case  that 
presented  itself.  It  is  preferable,  rather,  to  select 
a  few  simple,  familiar,  and  indubitable  theorems,  in 
our  choice  of  which  caprice  is  by  no  means  ex- 
cluded,1 and  to  formulate  from  these,  once  for  all, 
for  application  to  practical  ends,  general  proposi- 
tions answering  the  questions  that  most  frequently 
recur.  From  this  point  of  view  we  understand  at 
once  the  form  geometry  has  assumed, — the  empha- 
sis, for  example,  that  it  lays  upon  its  propositions 
concerning  triangles.  For  the  purpose  designated 
it  is  desirable  to  collect  the  most  general  possible 
propositions  having  the  widest  range  of  application. 
From  history  we  know  that  propositions  of  this 
character  have  been  obtained  by  embracing  various 
special  cases  of  knowledge  under  single  general 
cases.  We  are  forced  even  today  to  resort  to  this 
procedure  when  we  treat  the  relationship  of  two 
geometrical  figures,  or  when  the  different  special 
cases  of  form  and  position  compel  us  to  modify  our 
modes  of  deduction.  We  may  cite  as  the  most  fa- 
miliar instance  of  this  in  elementary  geometry,  the 


Windier.  Zur  Theorie  der  mathematischen  Erlcenntniss. 
Sitzungsberichte  der  Wiener  Akademie.  Philos-histor.  Abth. 
Bd.  118.  1889. 


92  SPACE    AND    GEOMETRY 

mode  of  deducing  the  relation  obtaining  between 
angles  at  the  centre  and  angles  at  the  circumference. 

UNIVERSAL  VALIDITY. 

Kroman1  has  put  the  question,  Why  do  we  regard 
a  demonstration  made  with  a  special  figure  (a  spe- 
cial triangle)  as  universally  valid  for  all  figures? 
and  finds  his  answer  in  the  supposition  that  we  are 
able  by  rapid  variations  to  impart  all  possible  forms 
to  the  figure  in  thought  and  so  convince  ourselves 
of  the  admissibility  of  the  same  mode  of  inference 
in  all  special  cases.  History  and  introspection  de- 
clare this  idea  to  be  in  all  essentials  correct.  But 
we  may  not  assume,  as  Kroman  does,  that  in  each 
special  case  every  individual  student  of  geometry 
acquires  this  perfect  comprehension  "with  the  rapid- 
ity of  lightning,"  and  reaches  immediately  the 
lucidity  and  intensity  of  geometric  conviction  in 
question.  Frequently  the  required  operation  is  abso- 
lutely impracticable,  and  errors  prove  that  in  other 
cases  it  was  actually  not  performed  but  that  the  in- 
quirer rested  content  with  a  conjecture  based  on 
analogy.2 

But  that  which  the  individual  does  not  or  cannot 
achieve  in  a  jiffy,  he  may  achieve  in  the  course  of 
his  life.  Whole  generations  labor  on  the  verifica- 
tion of  geometry.  And  the  conviction  of  its  certi- 
tude is  unquestionably  strengthened  by  their  collec- 

1  Unsere  Naturerkenntniss.    Copenhagen,  1883,  pp.  74  et  seq. 
'Hoelder,  Anschauung  und  Venken  in  der  Geometric,  p.  12. 


PSYCHOLOGY    AND    DEVELOPMENT    OF    GEOMETRY     93 

live  exertions.  I  once  knew  an  otherwise  excellent 
teacher  who  compelled  his  students  to  perform  all 
their  demonstrations  with  incorrect  figures,  on  the 
theory  that  it  was  the  logical  connection  of  the  con- 
cepts, not  the  figure,  that  was  essential.  But  the  ex- 
periences imbedded  in  the  concepts  cleave  to  our 
sensuous  images.  Only  the  actually  visualized  or 
imaged  figure  can  tell  us  what  particular  concepts 
are  to  be  employed  in  a  given  case.  The  method 
of  this  teacher  is  admirably  adapted  for  rendering 
palpable  the  degree  to  which  logical  operations  par- 
ticipate in  reaching  a  given  perception.  But  to  em- 
ploy it  habitually  is  to  miss  utterly  the  truth  that 
abstract  concepts  draw  their  ulitmate  power  from 
sensuous  sources. 


SPACE     AND     GEOMETRY     FROM     THE 

POINT  OF  VIEW  OF  PHYSICAL 

INQUIRY.1 

Our  notions  of  space  are  rooted  in  our  physiologi- 
cal organism.  Geometric  concepts  are  the  product 
of  the  idealization  of  physical  experiences  of  space. 
Systems  of  geometry,  finally,  originate  in  the  logical 
classification  of  the  conceptual  materials  so  obtained. 
All  three  factors  have  left  their  indubitable  traces  in 
modern  geometry.  Epistemological  inquiries  re- 
garding space  and  geometry  accordingly  concern 
the  physiologist,  the  psychologist,  the  physicist,  the 
mathematician,  the  philosopher,  and  the  logician 
alike,  and  they  can  be  gradually  carried  to  their 
definitive  solution  only  by  the  consideration  of  the 
widely  disparate  points  of  view  which  are  here  of- 
fered. 

Awakening  in  early  youth  to  full  consciousness, 
we  find  ourselves  in  possession  of  the  notion  of  a 
space  surrounding  and  encompassing  our  body,  in 
which  space  move  divers  bodies,  now  altering  and 


*I  shall  endeavor  in  this  essay  to  define  my  attitude  as  a 
physicist  toward  the  subject  of  metageometry  so  called.  De- 
tailed geometric  developments  will  have  to  be  sought  in  the 
sources.  I  trust,  however,  that  by  the  employment  of  illustra- 
tions which  are  familiar  to  every  one  I  have  made  my  exposi- 
tions as  popular  as  the  subject  permitted. 

94 


FROM  THE  POINT  OF  VIEW  OF  PHTSTCS  95 

now  retaining  their  size  and  shape.  It  is  impossible 
for  us  to  ascertain  how  this  notion  has  been  begot- 
ten. Only  the  most  thoroughgoing  analysis  of  ex- 
periments purposefully  and  methodically  performed 
has  enabled  us  to  conjecture  that  inborn  idiosyn- 
cracies  of  the  body  have  cooperated  to  this  end  with 
simple  and  crude  experiences  of  a  purely  physical 
character. 

SENSATIONAL  AND  LOCATIVE  QUALTIES. 

An  object  seen  or  touched  is  distinguished  not 
only  by  a  sensational  quality  (as  "red,"  "rough/' 
"cold,"  etc.),  but  also  by  a  locative  quality  (as  "to 
the  left,"  "above,"  "before,"  etc.).  The  sensational 
quality  may  remain  the  same,  while  the  locative 
quality  continuously  changes ;  that  is,  the  same  sen- 
suous object  may  move  in  space.  Phenomena  of  this 
kind  being  again  and  again  induced  by  physico-phys- 
ilogical  circumstances,  it  is  found  that  however  va- 
ried the  accidental  sensational  qualities  may  be,  the 
same  order  of  locative  qualities  invariably  occurs,  so 
that  the  latter  appear  perforce  as  a  fixed  and  perma- 
nent system  or  register  in  which  the  sensational 
qualities  are  entered  and  classified.  Now,  although 
these  qualities  of  sensation  and  locality  can  be  ex- 
cited only  in  conjunction  with  one  another,  and  can 
make  their  appearance  only  concomitantly,  the  im- 
pression nevertheless  easily  arises  that  the  more  fa- 
miliar system  of  locative  qualities  is  given  antece- 
dently to  the  sensational  qualities  (Kant). 

Extended  objects  of  vision  and  of  touch  consist 


96  SPACE    AND    GEOMETRY 

of  more  or  less  distinguishable  sensational  qualities, 
conjoined  with  adjacent  distinguishable,  contin- 
uously graduated  locative  qualities.  If  such  objects 
move,  particularly  in  the  domain  of  our  hands,  we 
perceive  them  to  shrink  or  swell  (in  whole  or  in 
part),  or  we  perceive  them  to  remain  the  same;  in 
other  words,  the  contrasts  characterizing  their 
bounding  locative  qualities  change  or  remain  con- 
stant. In  the  latter  case,  we  call  the  objects  rigid. 
By  the  recognition  of  permanency  as  coincident  with 
spatial  displacement,  the  various  constituents  of  our 
intuition  of  space  are  rendered  comparable  with  one 
another, — at  first  in  the  physiological  sense.  By  the 
comparison  of  different  bodies  with  one  another,  by 
the  introduction  of  physical  measures,  this  compar- 
ability is  rendered  quantitative  and  more  exact,  and 
so  transcends  the  limitations  of  individuality.  Thus, 
in  the  place  of  an  individual  and  non-transmittable 
intuition  of  space  are  substituted  the  universal  con- 
cepts of  geometry,  which  hold  good  for  all  men. 
Each  person  has  his  own  individual  intuitive  space; 
geometric  space  is  common  to  all.  Between  the 
space  of  intuition  and  metric  space,  which  contains 
physical  experiences,  we  must  distinguish  sharply. 

RIEMANN'S  PHYSICAL  CONCEPTION  OF  GEOMETRY. 

The  need  of  a  thoroughgoing  epistemological 
elucidation  of  the  foundations  of  geometry  induced 
Riemann,1  about  the  middle  of  the  century  just 

1  Ueber  die  Hypothesen,  welche  der  Geometric  zu   Grundc 
liegen.    Gottingen,  1867. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  97 

closed,  to  propound  the  question  of  the  nature  of 
space;  the  attention  of  Gauss,  Lobachevski,  and 
Bolyai  having  before  been  drawn  to  the  empirically 
hypothetical  character  of  certain  of  the  fundamental 
assumptions  of  geometry.  In  characterizing  space 
as  a  special  case  of  a  multiply-extended  "magni- 
tude," Riemann  had  doubtless  in  mind  some  geo- 
metric construct,  which  may  in  the  same  manner  be 
imagined  to  fill  all  space, — for  example,  the  system 
of  Cartesian  co-ordinates.  Riemann  further  asserts 
that  "the  propositions  of  geometry  cannot  be  deduced 
from  general  conceptions  of  magnitude,  but  that  the 
peculiar  properties  by  which  space  is  distinguished 
from  other  conceivable  triply-extended  magnitudes 
can  be  derived  from  experience  only ....  These 
facts,  like  all  facts,  are  in  no  wise  necessary,  but 
possess  empirical  certitude  only, — they  are  hypo- 
theses." Like  the  fundamental  assumptions  of 
every  natural  science,  so  also,  on  Riemann's  theory, 
the  fundamental  assumptions  of  geometry,  to  which 
experience  has  led  us,  are  merely  idealizations  of 
experience. 

In  this  physical  conception  of  geometry,  Riemann 
takes  his  stand  on  the  same  ground  as  his  master 
Gauss,  who  once  expressed  the  conviction  that  it 
was  impossible  to  establish  the  foundations  of 
geometry  entirely  a  priori*  and  who  further  as- 
serted that  "we  must  in  humility  confess  that  if 
number  is  exclusively  a  product  of  the  mind,  space 


1  Brief  von  Gauss  an  Bessel,  27.    Januar  1829. 


98  SPACE    AND    GEOMETRY 

possesses  in  addition  a  reality  outside  of  our  mind, 
of  which  reality  we  cannot  fully  dictate  a  priori  the 
laws."1 


ANALOGIES  OF  SPACE  WITH  COLORS. 

Every  inquirer  knows  that  the  knowledge  of  an 
object  he  is  investigating  is  materially  augmented 
by  comparing  it  with  related  objects.  Quite  natur- 
ally therefore  Riemann  looks  about  him  for  objects 
which  offer  some  analogy  to  space.  Geometric 
space  is  defined  by  him  as  a  triply-extended  contin- 
uous manifold,  the  elements  of  which  are  the  points 
determined  by  every  possible  three  co-ordinate  val- 
ues. He  finds  that  "the  places  of  sensuous  objects 
and  colors  are  probably  the  only  concepts  [sic] 
whose  modes  of  determination  form  a  multiply-ex- 
tended manifold."  To  this  analogy  others  were  add- 
ed by  Riemann' s  successors  and  elaborated  by  them, 
but  not  always,  I  think,  felicitously.2 

1  Brief  von  Gauss  an  Bessel.     April  9,  1830.— The  phrase, 
' '  Number  is  a  product  or  creation  of  the  mind, ' '  has  since 
been   repeatedly   used   by   mathematicians.     Unbiased   psycho- 
logical observation  informs  us,  however,  that  the  formation  of 
the  concept  of  number  is  just  as  much  initiated  by  experience 
as  the   formation  of  geometric  concepts.     We  must  at  least 
know  that  virtually  equivalent  objects  exist  in  multiple  and 
unalterable   form  before   concepts   of   number   can    originate. 
Experiments  in  counting  also  play  an  important  part  in  the  de- 
velopment of  arithmetic. 

2  When  acoustic  pitch,  intensity,  and  timbre,  when  chromatic 
tone,  saturation,  and  luminous  intensity  are  proposed   as  an- 
alogues of  the  three  dimensions  of  space,  few  persons  will  be 
satisfied.     Timbre,  like  chromatic  tone,  is  dependent  on  several 
variables.     Hence,  if  the  analogy  has  any  meaning  whatever, 
several  dimensions  will  be  found  to  correspond  to  timbre  and 
chromatic  tone. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  OX) 

Comparing  sensation  of  space  with  sensation  of 
color,  we  discover  that  to  the  continuous  series 
"above  and  below,"  "right  and  left,"  "near  and  far," 
correspond  the  three  sensational  series  of  mixed  col- 
ors, black-white,  red-green,  blue-yellow.  The  sys- 
tem of  sensed  (seen)  places  is  a  triple  continuous 
manifold  like  the  system  of  color-sensations.  The 
objection  which  is  raised  against  this  analogy,  viz., 
that  in  the  first  instance  the  three  variations  (di- 
mensions) are  homogeneous  and  interchangeable 
with  one  another,  while  in  the  second  instance  they 
are  heterogeneous  and  not  interchangeable,  does  not 
hold  when  space-sensation  is  compared  with  color- 
sensation.  For  from  the  psycho-physiological  point 
of  view  "right  and  left"  as  little  permit  of  being 
interchanged  with  "above  and  below"  as  do  red 
and  green  with  black  and  white.  It  is  only  when 
we  compare  geometric  space  with  the  system  of  col- 
ors that  the  objection  is  apparently  justified.  But 
there  is  still  a  great  deal  lacking  to  the  establish- 
ment of  a  complete  analogy  between  the  space  of  in- 
tuition and  the  system  of  color-sensation.  Whereas 
nearly  equal  distances  in  sensuous  space  are  imme- 
diately recognized  as  such,  a  like  remark  cannot  be 
made  of  differences  of  colors,  and  in  this  latter  prov- 
ince it  is  not  possible  to  compare  physiologically  the 
different  portions  with  one  another.  And,  further- 
more, even  if  there  be  no  difficulty,  by  resorting  to 
physical  experience,  in  characterizing  every  color  of 
a  system  by  three  numbers,  just  as  the  places  of 
geometric  space  are  characterized,  and  so  in  creat- 


IOO  SPACE    AND    GEOMETRY 

ing  a  metric  system  similar  to  the  latter,  it  will 
nevertheless  be  difficult  to  find  anything  which  cor- 
responds to  distance  or  volume  and  which  has  an 
analogous  physical  significance  for  the  system  of 
colors. 

ANALOGIES  OF  SPACE  WITH  TIME. 

There  is  always  an  arbitrary  element  in  analogies, 
for  they  are  concerned  with  the  coincidences  to 
which  the  attention  is  directed.  But  between  space 
and  time  doubtless  the  analogy  is  fully  conceded, 
whether  we  use  the  word  in  its  physiological  or  its 
physical  sense.  In  both  meanings  of  the  term,  space 
is  a  triple,  and  time  a  simple,  continuous  manifold. 
A  physical  event,  precisely  determined  by  its  condi- 
tions, of  moderate,  not  too  long  or  too  short  dura- 
tion, seems  to  us  physiologically,  now  and  at  any 
other  time,  as  having  the  same  duration.  Physical 
events  which  at  any  time  are  temporarily  coinci- 
dent are  likewise  temporarily  coincident  at  any  other 
time.  Temporal  congruence  exists,  therefore,  just 
as  much  as  does  spatial  congruence.  Unalterable 
physical  temporal  objects  exist,  therefore,  as  much 
as  unalterable  physical  spatial  objects  (rigid  bodies). 
There  is  not  only  spatial  but  there  is  also  temporal 
substantiality.  Galileo  employed  corporeal  phenom- 
ena, like  the  beats  of  the  pulse  and  breathing,  for  the 
determination  of  time,  just  as  anciently  the  hands 
and  the  feet  were  employed  for  the  estimation  of 
space. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  IOI 

The  simple  manifold  of  tonal  sensations  is  like- 
wise analogous  to  the  triple  manifold  of  space-sen- 
sations.1 The  comparability  of  the  different  parts 
of  the  system  of  tonal  sensations  is  given  by  the 
possibility  of  directly  sensing  the  musical  interval. 
A  metric  system  corresponding  to  geometric  space 
is  most  easily  obtained  by  expressing  tonal  pitch  in 
terms  of  the  logarithm  of  the  rate  of  vibration.  For 
the  constant  musical  interval  we  have  here  the  ex- 
pression, 

log  —  =  log  n'  —  log  n  =  log  T  —  log  T'  =  const., 

where  «',  n  denote  the  rates,  and  T',  r  the  periods  of 
vibration  of  the  higher  and  the  lower  note  respec- 
tively. The  difference  between  the  logarithms  here 
represents  the  constancy  of  the  length  on  displace- 
ment. The  unalterable,  substantial  physical  object 
which  we  sense  as  an  interval  is  for  the  ear  tempor- 
ally determined,  whereas  the  analogous  object  for 
the  senses  of  sight  and  touch  is  spatially  deter- 
mined. Spatial  measure  seems  to  us  simpler  solely 
because  we  have  chosen  for  the  fundamental  meas- 
ure of  geometry  distance  itself,  which  remains  un- 
alterable for  sensation,  whereas  in  the  province  of 
tones  we  have  reached  our  measure  only  by  a  long 
and  circuitous  physical  route. 


*My  attention  waa  drawn  to  this  analogy  in  1863  by  my 
study  of  the  organ  of  hearing,  and  I  have  since  then  further 
developed  the  subject.  See  my  Analysis  of  the  Sensations. 


IO2  SPACE    AND    GEOMETRY 

DIFFERENCES  OF  THE  ANALOGIES. 

Having  dwelt  on  the  coincidences  of  our  analo- 
gized constructs,  it  now  remains  for  us  to  emphasize 
their  differences.  Conceiving  time  and  space  as  sen- 
sational manifolds,  the  objects  whose  motions  are 
made  perceptible  by  the  alteration  of  temporal  and 
spatial  qualities  are  characterized  by  other  sensa- 
tional qualities,  as  colors,  tactual  sensations,  tones, 
etc.  If  the  system  of  tonal  sensations  is  regarded 
as  analogous  to  the  optical  space  of  sense,  the 
curious  fact  results  that  in  the  first  province  the 
spatial  qualities  occur  alone,  unaccompanied  by  sen- 
sational qualities  corresponding  to  the  objects,  just 
as  if  one  could  see  a  place  or  motion  without  seeing 
the  object  which  occupied  this  place  or  executed  this 
motion.  Conceiving  spatial  qualities  as  organic 
sensations  which  can  be  excited  only  concomitantly 
with  sensational  qualities,1  the  analogy  in  question 
does  not  appear  particularly  attractive.  For  the 
manifold-mathematician,  essentially  the  same  case 
is  presented  whether  an  object  of  definite  color 
moves  continuously  in  optical  space,  or  whether  an 
object  spatially  fixed  passes  continuously  through 
the  manifold  of  colors.  But  for  the  physiologist 
and  psychologist  the  two  cases  are  widely  different, 
not  only  because  of  what  was  above  adduced,  but 
also,  and  specifically,  because  of  the  fact  that  the 
system  of  spatial  qualities  is  very  familiar  to  us, 
whereas  we  can  represent  to  ourselves  a  system  of 

Compare  supra,  page  14  et  seq. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  IO3 

color-sensations  only  laboriously  and  artificially,  by 
means  of  scientific  devices.  Color  appears  to  us  as 
an  excerpted  member  of  a  manifold  the  arrange- 
ment of  which  is  in  no  wise  familiar  to  us. 

THE  EXTENSION  OF  SYMBOLS. 

The  manifolds  here  analogized  with  space  are, 
like  the  color-system,  also  threefold,  or  they  repre- 
sent a  smaller  number  of  variations.  Space  con- 
tains surfaces  as  twofold  and  lines  as  onefold  mani- 
folds, to  which  the  mathematician,  generalizing, 
might  also  add  points  as  zero-fold  manifolds.  There 
is  also  no  difficulty  in  conceiving  analytical  mechan- 
ics, with  Lagrange,  as  an  analytical  geometry  of 
four  dimensions,  time  being  considered  the  fourth 
co-ordinate.  In  fact,  the  equations  of  analytical 
geometry,  in  their  conformity  to  the  co-ordinates, 
suggest  very  clearly  to  the  mathematician  the  ex- 
tension of  these  considerations  to  an  unlimited 
larger  number  of  dimensions.  Similarly,  physics 
would  be  justified  in  considering  an  extended  mate- 
rial continuum,  to  each  point  of  which  a  tempera- 
ture, a  magnetic,  electric,  and  gravitational  poten- 
tial were  ascribed,  as  a  portion  or  section  of  a  multi- 
ple manifold.  Employment  with  such  symbolic 
representations  must,  as  the  history  of  science 
shows  us,  by  no  means  be  regarded  as  entirely  un- 
fruitful. Symbols  which  initially  appear  to  have  no 
meaning  whatever,  acquire  gradually,  after  subjec- 
tion to  what  might  be  called  intellectual  experi- 
menting, a  lucid  and  precise  significance.  Think 


IO4  SPACE    AND    GEOMETRY 

only  of  the  negative,  fractional,  and  variable  expo- 
nents of  algebra,  or  of  the  cases  in  which  important 
and  vital  extensions  of  ideas  have  taken  place  which 
otherwise  would  have  been  totally  lost  or  have  made 
their  appearance  at  a  much  later  date.  Think  only 
of  the  so-called  imaginary  quantities  with  which 
mathematicians  long  operated,  and  from  which  they 
even  obtained  important  results  ere  they  were  in  a 
position  to  assign  to  them  a  perfectly  determinate 
and  withal  visualizable  meaning.  But  symbolic  rep- 
resentation has  likewise  the  disadvantage  that  the 
object  represented  is  very  easily  lost  sight  of,  and 
that  operations  are  continued  with  the  symbols  to 
which  frequently  no  object  whatever  corresponds.1 

1  As  a  young  student  I  was  always  irritated  with  symbolic 
deductions  of  which  the  meaning  was  not  perfectly  clear  and 
palpable.  But  historical  studies  are  well  adapted  to  eradicat- 
ing the  tendency  to  mysticism  which  is  so  easily  fostered  and 
bred  by  the  somnolent  employment  of  these  methods,  in  that 
they  clearly  show  the  heuristic  function  of  them  and  at  the 
same  time  elucidate  epistemologically  the  points  wherein  they 
furnish  their  essential  assistance.  A  symbolical  representation 
of  a  method  of  calculation  has  the  same  significance  for  a 
mathematician  as  a  model  or  a  visualisable  working  hypothesis 
has  for  the  physicist.  The  symbol,  the  model,  the  hypothesis 
runs  parallel  with  the  thing  to  be  represented.  But  the  paral- 
lelism may  extend  farther,  or  be  extended  farther,  than  was 
originally  intended  on  the  adoption  of  the  symbol.  Since  the 
thing  represented  and  the  device  representing  are  after  all 
different,  what  would  be  concealed  in  the  one  is  apparent  in 
the  other.  It  is  scarcely  possible  to  light  directly  on  an  opera- 
tion like  03.  But  operating  with  such  symbols  leads  us  to 
attribute  to  them  an  intelligible  meaning.  Mathematicians 
worked  many  years  with  expressions  like  cos  x  X  V^ —  1  sin  « 
and  with  exponentials  having  imaginary  exponents  before 
in  the  struggle  for  adapting  concept  and  symbol  to  each  other 
the  idea  that  had  been  germinating  for  a  century  finally  found 
expression  in  1806  in  Argand,  viz.,  that  a  relationship  could  be 
conceived  between  magnitude  and  direction  by  which  V  —  1 
was  represented  as  a  mean  direction-proportional  between  +  1 
and  —  1. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  IO5 

ANOTHER  VIEW  OF  RIEMANN'S  MANIFOLD. 

It  is  easy  to  rise  to  Riemann's  conception  of  an 
n-fold  continuous  manifold,  and  it  is  even  possible 
to  realize  and  visualize  portions  of  such  a  manifold. 
Let  alf  a2,  a3,  a4  .  .  .  .  a  n+l  be  any  elements  whatso- 
ever (sensational  qualities,  substances,  etc.).  If  we 
conceive  these  elements  intermingled  in  all  their 
possible  relations,  then  each  single  composite  will  be 
represented  by  the  expression 

ai«i  +  a2^2  +  a3fla  +  ......  an+lan+l  =  I, 

where  the  coefficients  a  satisfy  the  equation 


Inasmuch,  therefore,  as  n  of  these  coefficients  a  may 
be  selected  at  pleasure,  the  totality  of  the  composites 
of  the  n  +  i  elements  will  represent  an  n-fold  con- 
tinuous manifold.1  As  co-ordinates  of  a  point  of 
this  manifold,  we  may  regard  expressions  of  the 
form 


•  or  ^r  for 

But  in  choosing  definition  of  distance,  or  that  of 
any  other  notion  analogous  to  geometrical  concepts, 
we  shall  have  to  proceed  very  arbitrarily  unless  ex- 
periences of  the  manifold  in  question  inform  us  that 
certain  metric  concepts  have  a  real  meaning,  and  are 
therefore  to  be  preferred,  as  is  the  case  for  geomet- 


*If  the  six  fundamental  color-sensations  were  totally  inde- 
pendent of  one  another,  the  system  of  color-sensations  would 
represent  a  five-fold  manifold.  Since  they  are  contrasted  in 
pairs,  the  system  corresponds  to  a  three-fold  manifold. 


106  SPACE    AND    GEOMETRY 

ric  space  with  the  definition1  derived  from  the  volum- 
inal  constancy  of  bodies  for  the  element  of  distances 
ds2  =  dx*  +  dyz  +  dz*,  and  as  is  likewise  the  case 
for  sensations  of  tone  with  the  logarithmic  expres- 
sion mentioned  above.  In  the  majority  of  cases 
where  such  an  artificial  construction  is  involved, 
fixed  points  of  this  sort  are  wanting1,  and  the  entire 
consideration  is  therefore  an  ideal  one.  The  anal- 
ogy with  space  loses  thereby  in  completeness,  fruit- 
fulness,  and  stimulating  power. 

MEASURE  OF  CURVATURE,  AND  CURVATURE  OF 
SPACE. 

In  still  another  direction  Riemann  elaborated 
ideas  of  Gauss;  beginning  with  the  latter's  investi- 
gations concerning  curved  surfaces.  Gauss's  meas- 
ure of  the  curvature2  of  a  surface  at  any  point  is 

given  by  the  expression  k  =  —    where  ds  is  an  ele- 

ment of  the  surface  and  d<r  is  the  superficial  element 
of  the  unit-sphere,  the  limiting  radii  of  which  are 
parallel  to  the  limiting  normals  of  the  element  ds. 
This  measure  of  curvature  may  also  be  expressed  in 

the  form  k  =  —  l—  ,     where  Pi»f>2  are  the  principal 

' 


radii  of  curvature  of  the  surface  at  the  point  in 
question.  Of  special  interest  are  the  surfaces  whose 
measure  of  curvature  for  all  points  has  the  same 


1  Comp.  supra,  p.  73  et  passim. 

8  Disquisit iones  generates  circa  superficies  curvas,  1827. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  IO/ 

value, — the  surfaces  of  constant  curvature.  Con- 
ceiving the  surfaces  as  infinitely  thin,  non-distensi- 
ble, but  flexible  bodies,  it  will  be  found  that  sur- 
faces of  like  curvature  may  be  made  to  coincide  by 
bending, — as  for  example  a  plane  sheet  of  paper 
wrapped  round  a  cylinder  or  cone, — but  cannot  be 
made  to  coincide  with  the  surface  of  a  sphere.  Dur- 
ing such  deformation,  nay,  even  on  crumpling,  the 
proportional  parts  of  figures  drawn  in  the  surface 
remain  invariable  as  to  lengths  and  angles,  provided 
we  do  not  go  out  of  the  two  dimensions  of  the  sur- 
face in  our  measurements.  Conversely,  likewise, 
the  curvature  of  the  surface  does  not  depend  on  its 
conformation  in  the  third  dimension  of  space,  but 
solely  upon  its  interior  proportionalities.  Riemann, 
now,  conceived  the  idea  of  generalizing  the  notion 
of  measure  of  curvature  and  applying  it  to  spaces 
of  three  or  more  dimensions.  Conformably  there- 
to, he  assumes  that  finite  unbounded  spaces  of  con- 
stant positive  curvature  are  possible,  corresponding 
to  the  unbounded  but  finite  two-dimensional  surface 
of  the  sphere,  while  what  we  commonly  take  to  be 
infinite  space  would  correspond  to  the  unlimited 
plane  of  curvature  zero,  and  similarly  a  third  spe- 
cies of  space  would  correspond  to  surfaces  of  neg- 
ative curvature.  Just  as  the  figures  drawn  upon  a 
surface  of  determinate  constant  curvature  can  be 
displaced  without  distortion  upon  this  surface  only 
(for  example,  a  spherical  figure  on  the  surface  of 
its  sphere  only,  or  a  plane  figure  in  its  plane  only), 
so  should  analogous  conditions  necessarily  hold  for 


IO8  SPACE    AND    GEOMETRY 

spatial  figures  and  rigid  bodies.  The  latter  are 
capable  of  free  motion  only  in  spaces  of  constant 
curvature,  as  Helmholtz1  has  shown  at  length.  Just 
as  the  shortest  lines  of  a  plane  are  infinite,  but  on 
the  surface  of  a  sphere  occur  as  great  circles  of  defi- 
nite finite  length,  closed  and  reverting  into  them- 
selves, so  Riemann  conceived  in  the  three-dimen- 
sional space  of  positive  curvature  analogues  of  the 
straight  line  and  the  plane  as  finite  but  unbounded. 
But  there  is  a  difficulty  here.  If  we  possessed  the 
notion  of  a  measure  of  curvature  for  a  four-dimen- 
sional space,  the  transition  to  the  special  case  of 
three-dimensional  space  could  be  easily  and  ration- 
ally executed;  but  the  passage  from  the  special  to 
the  more  general  case  involves  a  certain  arbitrari- 
ness, and,  as  is  natural,  different  inquirers  have 
adopted  here  different  courses2  (Riemann  and  Kro- 
necker).  The  very  fact  that  for  a  one-dimensional 
space  (a  curved  line  of  any  sort)  a  measure  of  curv- 
ature does  not  exist  having  the  significance  of  an  in- 
terior measure,  and  that  such  a  measure  first  occurs 
in  connection  with  two-dimensional  figures,  forces 
upon  us  the  question  whether  and  to  what  extent 
something  analogous  has  any  meaning  for  three- 
dimensional  figures.  Are  we  not  subject  here  to  an 
illusion,  in  that  we  operate  with  symbols  to  which 
perhaps  nothing  real  corresponds,  or  at  least  noth- 

"'Ueber  die  Thatsachen,  welche  der  Geometrie  zu  Grunde 
liegen. ' '  Gottinger  Nachrichten,  1868,  June  3. 

2  Compare,  for  example,  Kronecker,  ' '  Ueber  Systeme  von 
Functionen  mehrerer  Variablen."  Ber.  d.  Berliner  Akademie, 
1869. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  IOO, 

ing  representable  to  the  senses,  by  means  of  which 
we  can  verify  and  rectify  our  ideas? 

Thus  were  reached  the  highest  and  most  univer- 
sal notions  regarding  space  and  its  relations  to 
analogous  manifolds  which  resulted  from  the  con- 
viction of  Gauss  concerning  the  empirical  founda- 
tions of  geometry.  But  the  genesis  of  this  convic- 
tion has  a  preliminary  history  of  two  thousand 
years,  the  chief  phenomena  of  which  we  can  perhaps 
better  survey  from  the  height  which  we  have  now 
gained. 

THE  EARLY  DISCOVERIES  IN  GEOMETRY. 

The  unsophisticated  men,  who,  rule  in  hand,  ac- 
quired our  first  geometric  knowledge,  held  to  the 
simplest  bodily  objects  (figures)  :  the  straight  line, 
the  plane,  the  circle,  etc.,  and  investigated,  by  means 
of  forms  which  could  be  conceived  as  combinations 
of  these  simple  figures,  the  connection  of  their 
measurements.  It  could  not  have  escaped  them  that 
the  mobility  of  a  body  is  restricted  when  one  and 
then  two  of  its  points  are  fixed,  and  that  finally  it 
is  altogether  checked  by  fixing  three  of  its  points. 
Granting  that  rotation  about  an  axis  (two  points), 
or  rotation  about  a  point  in  a  plane,  as  likewise  dis- 
placement with  constant  contact  of  two  points  with 
a  straight  line  and  of  a  third  point  with  a  fixed 
plane  laid  through  that  straight  line, — granting  that 
these  facts  were  separately  observed,  it  would  be 
known  how  to  distinguish  between  pure  rotation, 


no 


SPACE    AND    GEOMETRY 


pure  displacement,  and  the  motion  compounded  of 
these  two  independent  motions.  The  first  geometry 
was  of  course  not  based  on  purely  metric  notions, 
but  made  many  considerable  concessions  to  the  phy- 
siological factors  of  sense.1  Thus  is  the  appearance 
explained  of  two  different  fundamental  measures: 
the  (straight)  length  and  the  angle  (circular  meas- 
ure). The  straight  line  was  conceived  as  a  rigid 
mobile  body  (measuring- rod),  and  the  angle  as  the 


Pig.   14. 

rotation  of  a  straight  line  with  respect  to  another 
(measured  by  the  arc  so  described).  Doubtless  no 
one  ever  demanded  special  proof  for  the  equality  of 
angles  at  the  origin  described  by  the  same  rotation. 
Additional  propositions  concerning  angles  resulted 
quite  easily.  Turning  the  line  b  about  its  intersec- 
tion with  c  so  as  to  describe  the  angle  a  (Fig.  14), 
and  after  coincidence  with  c  turning  it  again  about 


.  supra,  p.  83. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS 


III 


its  intersection  with  a  till  it  coincides  with  a  and  so 
describes  the  angle  /?,  we  shall  have  rotated  b  from 
its  initial  to  its  final  position  a  through  the  angle  /* 
in  the  same  sense.1  Therefore  the  exterior  angle  /* 
=  a  -f-  /?,  and  since  /*+  y  =  zRf  also  a  +  ft  +  y  = 
2R.  Displacing  (Fig.  15)  the  rigid  system  of  lines 
a,  b,  c,  which  intersect  at  i,  within  their  plane  to  the 
position  2,  the  line  a  always  remaining  within  itself, 
no  alteration  of  angles  will  be  caused  by  the  mere 


Fig.  15. 


motion.  The  sum  of  the  interior  angles  of  the  tri- 
angle i  2  3  so  produced  is  evidently  2  R.  The  same 
consideration  also  throws  into  relief  the  properties 


1C.  R.  Kosack,  Beitrage  zu  einer  systematischen  Entwickel- 
ung  der  Geometric  aus  der  Anschauung,  Nordhausen,  1852.  I 
was  able  to  see  this  programme  through  the  kindness  of  Prof. 
F.  Pietzker  of  Nordhausen.  Similar  simple  deductions  are 
found  in  Bernhard  Becker's  Leitfaden  fur  den  ersten  Unter- 
richt  in  der  Geometric,  Frankfort  on  the  Main,  1845,  and  in 
the  same  author's  treatise  Ueber  die  Methoden  des  geo- 
metrischen  Unterrichts,  Frankfort,  1845.  I  gained  access  to 
the  first-named  book  through  the  kindness  of  Dr.  M.  Schuster 
of  Oldenburg. 


112  SPACE    AND    GEOMETRY 

of  parallel  lines.  Doubts  as  to  whether  successive 
rotation  about  several  points  is  equivalent  to  rota- 
tion about  one  point,  whether  pure  displacement  is 
at  all  possible, — which  are  justified  when  a  surface 
of  curvature  differing  from  zero  is  substituted  for 
the  Euclidean  plane, — could  never  have  arisen  in 
the  mind  of  the  ingenuous  and  delighted  discoverer 
of  these  relations,  at  the  period  we  are  considering. 
The  study  of  the  movement  of  rigid  bodies,  which 
Euclid  studiously  avoids  and  only  covertly  intro- 
duces in  his  principle  of  congruence,  is  to  this  day 
the  device  best  adapted  to  elementary  instruction  in 
geometry.  An  idea  is  best  made  the  possession  of 
the  learner  by  the  method  by  which  it  has  been 
found. 

DEDUCTIVE  GEOMETRY. 

This  sound  and  naive  conception  of  things  van- 
ished and  the  treatment  of  geometry  underwent  es- 
sential modifications  when  it  became  the  subject  of 
professional  and  scholarly  contemplation.  The  ob- 
ject now  was  to  systematize  the  knowledge  of  this 
province  for  purposes  of  individual  survey,  to  sepa- 
rate what  was  directly  cognizable  from  what  was 
deducible  and  deduced,  and  to  throw  into  distinct 
relief  the  thread  of  deduction.  For  the  purpose  of 
instruction  the  simplest  principles,  those  most  easily 
gained  and  apparently  free  from  doubt  and  contra- 
diction, are  placed  at  the  beginning,  and  the  remain- 
der based  upon  them.  Efforts  were  made  to  reduce 


FROM   THE   POINT  OF  VIEW  OF  PHYSICS  113 

these  initial  principles  to  a  minimum,  as  is  observ- 
able in  the  system  of  Euclid.  Through  this  en- 
deavor to  support  every  notion  by  another,  and  to 
leave  to  direct  knowledge  the  least  possible  scope, 
geometry  was  gradually  detached  from  the  empiri- 
cal soil  out  of  which  it  had  sprung.  People  accus- 
tomed themselves  to  regard  the  derived  truths  as  of 
higher  dignity  than  the  directly  perceived  truths, 
and  ultimately  came  to  demand  proofs  for  proposi- 
tions which  no  one  ever  seriously  doubted.  Thus 
arose, — as  tradition  would  have  it,  to  check  the  on- 
slaughts of  the  Sophists, — the  system  of  Euclid  with 
its  logical  perfection  and  finish.  Yet  not  only  were 
the  ways  of  research  designedly  concealed  by  this 
artificial  method  of  stringing  propositions  on  an 
arbitrarily  chosen  thread  of  deduction,  but  the  var- 
ied organic  connection  between  the  principles  of 
geometry  was  quite  lost  sight  of.1  This  system  was 
more  fitted  to  produce  narrow-minded  and  sterile 
pedants  than  fruitful,  productive  investigators. 


Euclid's  system  fascinated  thinkers  by  its  logical  excel- 
lences, and  its  drawbacks  were  overlooked  amid  this  admiration. 
Great  inquirers,  even  in  recent  times,  hare  been  misled  into 
following  Euclid's  example  in  the  presentation  of  the  results 
of  their  inquiries,  and  so  into  actually  concealing  their  methods 
of  investigation,  to  the  great  detriment  of  science.  But  sci- 
ence is  not  a  feat  of  legal  casuistry.  Scientific  presentation 
aims  so  to  expound  all  the  grounds  of  an  idea  so  that  it  can 
at  any  time  be  thoroughly  examined  as  to  its  tenability  and 
power.  The  learner  is  not  to  be  led  half-blindfolded.  There 
therefore  arose  in  Germany  among  philosophers  and  education- 
ists a  healthy  reaction,  which  proceeded  mainly  from  Herbart, 
Schopenhauer,  and  Trendelenburg.  The  effort  was  made  to 
introduce  greater  perspicuity,  more  genetic  methods,  and  logi- 
cally more  lucid  demonstrations  into  geometry. 


114  SPACE    AND    GEOMETRY 

And  these  conditions  were  not  improved  when 
scholasticism,  with  its  preference  for  slavish  com- 
ment on  the  intellectual  products  of  others,  culti- 
vated in  thinkers  scarcely  any  sensitiveness  for  the 
rationality  of  their  fundamental  assumptions  and 
by  way  of  compensation  fostered  in  them  an  exag- 
gerated respect  for  the  logical  form  of  their  deduc- 
tions. The  entire  period  from  Euclid  to  Gauss  suf- 
fered more  or  less  from  this  affection  of  mind. 

EUCLID'S  FIFTH  POSTULATE. 

Among  the  propositions  on  which  Euclid  based 
his  system  is  found  the  so-called  Fifth  Postulate 
(also  called  the  Eleventh  Axiom  and  by  some  the 
Twelfth)  :  "If  a  straight  line  meet  two  straight 
lines,  so  as  to  make  the  two  interior  angles  on  the 
same  side  of  it  taken  together  less  than  two  right 
angles,  these  straight  lines  being  continually  pro- 
duced, shall  at  length  meet  upon  that  side  on  which 
are  the  angles  which  are  less  than  two  right  an- 
gles." Euclid  easily  proves  that  if  a  straight  line 
falling  on  two  other  straight  lines  makes  the  alter- 
nate angles  equal  to  each  other,  the  two  straight 
lines  will  not  meet  but  are  parallel.  But  for  the 
proof  of  the  converse,  that  parallels  make  equal 
alternate  angles  with  every  straight  line  falling  on 
them,  he  is  obliged  to  resort  to  the  Fifth  Postulate. 
This  converse  is  equivalent  to  the  proposition  that 
only  one  parallel  to  a  straight  line  can  be  drawn 
through  a  point.  Further,  by  the  fact  that  with  the 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  11$ 

aid  of  this  converse  it  can  be  proved  that  the  sum 
of  the  angles  of  a  triangle  is  equal  to  two  right  an- 
gles and  that  from  this  last  theorem  again  the  first 
follows,  the  relationship  between  the  propositions  in 
question  is  rendered  distinct  and  the  fundamental 
significance  of  the  Fifth  Postulate  for  Euclidean 
geometry  is  made  plain. 

The  intersection  of  slowly  converging  lines  lies 
without  the  province  of  construction  and  observa- 
tion. It  is  therefore  intelligible  that  in  view  of  the 
great  importance  of  the  assertion  contained  in  the 
Fifth  Postulate  the  successors  of  Euclid,  habituated 
by  him  to  rigor,  should,  even  in  ancient  times,  have 
strained  every  nerve  to  demonstrate  this  postulate, 
or  to  replace  it  by  some  immediately  obvious  propo- 
sition. Numberless  futile  efforts  were  made  from 
Euclid  to  Gauss,  to  deduce  this  Fifth  Postulate  from 
the  other  Euclidean  assumptions.  It  is  a  sublime 
spectacle  which  these  men  offer:  laboring  for  cen- 
turies, from  a  sheer  thirst  for  scientific  elucidation, 
in  quest  of  the  hidden  sources  of  a  truth  which  no 
person  of  theory  or  of  practice  ever  really  doubted! 
With  eager  curiosity  we  follow  the  pertinacious  ut- 
terances of  the  ethical  power  resident  in  this  human 
search  for  knowledge,  and  with  gratification  we 
note  how  the  inquirers  gradually  are  led  by  their 
failures  to  the  perception  that  the  true  basis  of 
geometry  is  experience.  We  shall  content  ourselves 
with  a  few  examples. 


Il6  SPACE  AND  GEOMETRY 

SACCHERI'S  THEORY  OF  PARALLELS. 

Among  the  inquirers  notable  for  their  contribu- 
tions to  the  theory  of  parallels  are  the  Italian  Sac- 
cheri  and  the  German  mathematician  Lambert.  In 
order  to  render  their  mode  of  attack  intelligible,  we 
will  remark  first  that  the  existence  of  rectangles  and 
squares,  which  we  fancy  we  constantly  observe,  can- 
not be  demonstrated  without  the  aid  of  the  Fifth 
Postulate.  Let  us  consider,  for  example,  two  con- 
gruent isosceles  triangles  ABC,  DBC,  having  right 
angles  at  A  and  D  (Fig.  16),  and  let  them  be  laid 
together  at  their  hypothenuses  BC  so  as  to  form  the 

B 


Fig.  16. 

equilateral  quadrilateral  ABCD;  the  first  twenty- 
seven  propositions  of  Euclid  do  not  suffice  to  deter- 
mine the  character  and  magnitude  of  the  two  equal 
(right)  angles  at  B  and  C.  For  measure  of  length 
and  measure  of  angle  are  fundamentally  different 
and  directly  not  comparable;  hence  the  first  propo- 
sitions regarding  the  connection  of  sides  and  angles 
are  qualitative  only,  and  hence  the  imperative  neces- 
sity of  a  quantitative  theorem  regarding  angles,  like 
that  of  the  angle-sum.  Be  it  further  remarked  that 
theorems  analogous  to  the  twenty-seven  planimetric 
propositions  of  Euclid  may  be  set  up  for  the  surface 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS 


117 


of  a  sphere  and  for  surfaces  of  constant  negative 
curvature,  and  that  in  these  cases  the  analogous 
construction  gives  respectively  obtuse  and  acute  an- 
gles at  B  and  C. 

Saccheri's  cardinal  achievement  was  his  form  of 
stating  the  problem.1  If  the  Fifth  Postulate  is  in- 
volved in  the  remaining  assumptions  of  Euclid, 
then  it  will  be  possible  to  prove  without  its  aid  that 
in  the  quadrilateral  ABCD  (Fig.  17)  having  right 
angles  at  A  and  B  and  AC  =  BD,  the  angles  at  C 
and  D  likewise  are  right  angles.  And,  on  the  other 
hand,  in  this  event,  the  assumption  that  C  and  D 


M 
Fig.  17. 

are  either  obtuse  or  acute  will  lead  to  contradictions. 
Saccheri,  in  other  words,  seeks  to  draw  conclusions 
from  the  hypothesis  of  the  right,  the  obtuse,  or  the 
acute  angle.  He  shows  that  each  of  these  hypothe- 
ses will  hold  in  all  cases  if  it  be  proved  to  hold  in 
one.  It  is  needful  to  have  only  one  triangle  with 
its  angles  =  2R  in  order  to  demonstrate  the  univer- 
sal validity  of  the  hypothesis  of  the  acute,  the  right, 
or  the  obtuse  angle.  Notable  is  the  fact  that  Sac- 
cheri also  adverts  to  physico-geometrical  experi- 

1  Euclides  ab  omni  naevo  vindicates.  Milan,  1733.  German 
translation  in  Engel  and  Staeckel's  Die  Theorie  der  Parallel- 
linien.  Leipsic,  1895. 


Il8  SPACE    AND    GEOMETRY 

ments  which  support  the  hypothesis  of  the  right 
angle.  If  a  line  CD  (Fig.  17)  join  the  two  extremi- 
ties of  the  equal  perpendiculars  erected  on  a  straight 
line  AB,  and  the  perpendicular  dropped  on  AB  from 
any  point  N  of  the  first  line,  viz.,  JVM,  be  equal  to 
CA  =  DB,  then  is  the  hypothesis  of  the  right  angle 
demonstrated  to  be  correct.  Saccheri  rightly  does 
not  regard  it  as  self-evident  that  the  line  which  is 
equidistant  from  another  straight  line  is  itself  a 
straight  line.  Think  only  of  a  circle  parallel  to  a 
great  circle  on  a  sphere  which  does  not  represent  a 


Fig.  18.  Fig.  19. 

shortest  line  on  a  sphere  and  the  two  faces  of  which 
cannot  be  made  congruent. 

Other  experimental  proofs  of  the  correctness  of 
the  hypothesis  of  the  right  angle  are  the  following. 
If  the  angle  in  a  semicircle  (Fig.  18)  is  shown  to  be 
a  right  angle,  a  +  /?  =  Rf  then  is  2a  +  2/?  =  2R, 
the  sum  of  the  angles  of  the  triangle  ABC.  If  the 
radius  be  subtended  thrice  in  a  semicircle  and  the 
line  joining  the  first  and  the  fourth  extremity  pass 
through  the  center,  we  shall  have  at  C  (Fig.  19) 
£a  =  2R,  and  consequently  each  of  the  three  tri- 
angles will  have  the  angle-sum  2.R.  The  existence 
of  equiangular  triangles  of  different  sizes  (similar 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS 


119 


triangles)  is  likewise  subject  to  experimental  proof. 
For  (Fig.  20)  if  the  angles  at  B  and  C  give  /?  +  8  + 
y  +  « =  4~ft,  so  also  is  4^  the  angle-sum  of  the 
quadrilateral  BCB'C'.  Even  Wallis1  (1663)  based 
his  proof  of  the  Fifth  Postulate  on  the  assumption 
of  the  existence  of  similar  triangles,  and  a  modern 
geometer,  Delbceuf,  deduced  from  the  assumption 
of  similitude  the  entire  Euclidean  geometry. 
The  hypothesis  of  the  obtuse  angle,  Saccheri  fan- 


Fig.  20. 

cied  he  could  easily  refute.  But  the  hypothesis  of 
the  acute  angle  presented  to  him  difficulties,  and  in 
his  quest  for  the  expected  contradictions  he  was  car- 
ried to  the  most  far-reaching  conclusions,  which 
Lobachevski  and  Bolyai  subsequently  rediscovered 
by  methods  of  their  own.  Ultimately  he  felt  com- 
pelled to  reject  the  last-named  hypothesis  as  incom- 
patible with  the  nature  of  the  straight  line;  for  it 

1  Engel  and  Staeckel,  loc.  eit.,  p.  21  et  seq. 


120  SPACE    AND    GEOMETRY 

led  to  the  assumption  of  different  kinds  of  straight 
lines,  which  met  at  infinity,  that  is,  had  there  a  com- 
mon perpendicular.  Saccheri  did  much  in  anticipa- 
tion and  promotion  of  the  labors  that  were  subse- 
quently to  elucidate  these  matters,  but  exhibited 
withal  toward  the  traditional  views  a  certain  bias. 

LAMBERT'S  INVESTIGATIONS. 

Lambert's  treatise1  is  allied  in  method  to  that  of 
Saccheri,  but  it  proceeds  farther  in  its  conclusions, 
and  gives  evidence  of  a  less  constrained  vision. 
Lambert  starts  from  the  consideration  of  a  quadri- 
lateral with  three  right  angles,  and  examines  the 
consequences  that  would  follow  from  the  assumption 
that  the  fourth  angle  was  right,  obtuse,  or  acute. 
The  similarity  of  figures  he  finds  to  be  incompatible 
with  the  second  and  third  assumptions.  The  case  of 
the  obtuse  angle,  which  requires  the  sum  of  the  an- 
gles of  a  triangle  to  exceed  2R,  he  discovers  to  be 
realized  in  the  geometry  of  spherical  surfaces,  in 
which  the  difficulty  of  parallel  lines  entirely  van- 
ishes. This  leads  him  to  the  conjecture  that  the 
case  of  the  acute  angle,  where  the  sum  of  the  angles 
of  a  triangle  is  less  than  2R,  might  be  realized  on 
the  surface  of  a  sphere  of  imaginary  radius.  The 
amount  of  the  departure  of  the  angle-sum  from  2R 
is  in  both  cases  proportional  to  the  area  of  the  tri- 
angle, as  may  be  demonstrated  by  appropriately  di- 


'Published  in  1766.     Engel  and  Staeckel,  loc  cit.,  p.  152  et 
eeq. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  121 

viding  large  triangles  into  small  triangles,  which  on 
diminution  may  be  made  to  approach  as  near  as  w$ 
please  to  the  angle-sum  2R.  Lambert  advanced 
very  closely  in  this  conception  to  the  point  of  view 
of  modern  geometers.  Admittedly  a  sphere  of  im- 
aginary radius,  rV  —  i  is  not  a  visualizable  geo- 
metric construct,  but  analytically  it  is  a  surface  hav- 
ing a  negative  constant  Gaussian  measure  of  curva- 
ture. It  is  evident  again  from  this  example  how 
experimenting  with  symbols  also  may  direct  inquiry 
to  the  right  path,  in  periods  where  other  points  of 
support  are  entirely  lacking  and  where  every  help- 
ful device  must  be  esteemed  at  its  worth.1  Even 
Gauss  appears  to  have  thought  of  a  sphere  of  im- 
aginary radius,  as  is  obvious  from  his  formula  for 
the  circumference  of  a  circle  (Letter  to  Schumacher, 
July  12,  1831).  Yet  in  spite  of  all,  Lambert  actu- 
ally fancied  he  had  approached  so  near  to  the  proof 
of  the  Fifth  Postulate  that  what  was  lacking  could 
be  easily  supplied. 

VIEW  OF  GAUSS. 

We  may  turn  now  to  the  investigators  whose 
views  possess  a  most  radical  significance  for  our 
conception  of  geometry,  but  who  announced  their 
opinion  only  briefly,  by  word  of  mouth  or  letter. 
"Gauss  regarded  geometry  merely  as  a  logically  con- 
sistent system  of  constructs,  with  the  theory  of  par- 
allels placed  at  the  pinnacle  as  an  axiom ;  yet  he  had 

1  See  note,  p.  104. 


122 


SPACE    AND    GEOMETRY 


reached  the  conviction  that  this  proposition  could 
not  be  proved,  though  it  was  known  from  experi- 
ence,— for  example,  from  the  angles  of  the  triangle 
joining  the  Brocken,  Hohenhagen,  and  Inselsberg, 
— that  it  was  approximately  correct.  But  if  this 
axiom  be  not  conceded,  then,  he  contends,  there  re- 
sults from  its  non-acceptance  a  different  and  entirely 
independent  geometry,  which  he  had  once  investi- 
gated and  called  by  the  name  of  the  Anti-Euclidean 


Fig.  21. 

geometry."     Such,  according  to  Sartorius  von  Wal- 
tershausen,  was  the  view  of  Gauss.1 


RESEARCHES  OF  STOLZ. 

Starting  at  this  point,  O.  Stolz,  in  a  small  but 
very  instructive  pamphlet,2  sought  to  deduce  the 
principal  propositions  of  the  Euclidean  geometry 
from  the  purely  observable  facts  of  experience.  We 
shall  reproduce  here  the  most  important  point  of 
Stolz's  brochure.  Let  there  be  given  (Fig.  21)  one 


1  Gauss  zum  Gedachtniss,  Leipsic,  1856. 

'"Das  letzte  Axiom  der  Geometric,"  Berirfite  des  naturw.- 
medicin.  Vereim  zu  Innslruclc,  1886,  pp.  25-34. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS 


I23 


large  triangle  ABC  having  the  angle-sum  2R.  We 
draw  the  perpendicular  AD  on  BC,  complete  the 
figure  by  BAR  &  ABD  and  CAP  £  ACD,  and  add 
to  the  figure  BCFAE  the  congruent  figure  CBHA'G. 
We  obtain  thus  a  single  rectangle,  for  the  angles 
E,  F ,  G,  H  are  right  angles  and  those  at  A,  C,  A' , 
B  are  straight  angles  (equal  to  2R),  the  boundary 
lines  therefore  straight  lines  and  the  opposite  sides 
equal.  A  rectangle  can  be  divided  into  two  congru- 
ent rectangles  by  a  perpendicular  erected  at  the 
middle  point  of  one  of  its  sides,  and  by  continuing 
this  procedure  the  line  of  division  may  be  brought 


B 


Fig.  22. 


to  any  point  we  please  in  the  divided  side.  And  the 
same  holds  true  of  the  other  two  opposite  sides.  It 
is  possible,  therefore,  from  a  given  rectangle  ABCD 
(Fig.  22)  to  cut  out  a  smaller  AMPQ  having  sides 
bearing  any  proportion  to  one  another.  The  diag- 
onal of  this  last  divides  it  into  two  congruent  right- 
angled  triangles,  of  which  each,  independently  of 
the  ratio  of  the  sides,  has  the  angle-sum  2.R.  Every 
oblique-angled  triangle  can  by  the  drawing  of  a  per- 
pendicular be  decomposed  into  right-angled  trian- 
gles, each  of  which  can  again  be  decomposed  into 


124  SPACE    AND    GEOMETRY 

right-angled  triangles  having  smaller  sides, — so 
that  2Rf  therefore,  results  for  the  angle-sum  of 
every  triangle  if  it  holds  true  exactly  of  one.  By  the 
aid  of  these  propositions  which  repose  on  observa- 
tion we  conclude  easily  that  the  two  opposite  sides 
of  a  rectangle  (or  of  any  so-called  parallelogram) 
are  everywhere,  no  matter  how  far  prolonged,  the 
same  distance  apart,  that  is,  never  intersect.  They 
have  the  properties  of  the  Euclidean  parallels,  and 
may  be  called  and  defined  as  such.  It  likewise  fol- 
lows, now,  from  the  properties  of  triangles  and  rect- 
angles, that  two  straight  lines  which  are  cut  by  a 
third  straight  line  so  as  to  make  the  sum  of  the  in- 
terior angles  on  the  same  side  of  them  less  than  two 
right  angles  will  meet  on  that  side,  but  in  either 
direction  from  their  point  of  intersection  will  move 
indefinitely  far  away  from  each  other.  The  straight 
line  therefore  is  infinite.  What  was  a  groundless 
assertion  stated  as  an  axiom  or  an  initial  principle 
may  as  inference  have  a  sound  meaning. 

GEOMETRY  AND  PHYSICS  COMPARED. 

Geometry,  accordingly,  consists  of  the  application 
of  mathematics  to  experiences  concerning  space. 
Like  mathematical  physics,  it  can  become  an  exact 
deductive  science  only  on  the  condition  of  its  repre- 
senting the  objects  of  experience  by  means  of 
schematizing  and  idealizing  concepts.  Just  as  me- 
chanics can  assert  the  constancy  of  masses  or  reduce 
the  interactions  between  bodies  to  simple  accelera- 
tions only  within  the  limits  of  errors  of  observation, 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  125 

so  likewise  the  existence  of  straight  lines,  planes,  the 
amount  of  the  angle-sum,  etc.,  can  be  maintained 
only  on  a  similar  restriction.  But  just  as  physics 
sometimes  finds  itself  constrained  to  replace  its  ideal 
assumptions  by  other  more  general  ones,  viz.,  to  put 
in  the  place  of  a  constant  acceleration  of  falling  bod- 
ies one  dependent  on  the  distance,  instead  of  a  con- 
stant quantity  of  heat  a  variable  quantity, — so  a 
similar  procedure  is  permissible  in  geometry,  when 
it  is  demanded  by  the  facts  or  is  necessary  tempor- 
arily for  scientific  elucidation.  And  now  the  en- 
deavors of  Legendre,  Lobachevski,  and  the  two 
Bolyais,  the  younger  of  whom  was  probably  indi- 
rectly inspired  by  Gauss,  will  appear  in  their  right 
light. 

THE  CONTRIBUTIONS  OF  LOBACHEVSKI  AND 
BOLYAI. 

Of  the  labors  of  Schweickart  and  Taurinus,  also 
contemporaries  of  Gauss,  we  will  not  speak.  Lo- 
bachevski's  works  were  the  first  to  become  known 
to  the  thinking  world  and  so  productive  of  results 
(1829).  Very  soon  afterward  the  publication  of 
the  younger  Bolyai  appeared  (1833),  which  agreed 
in  all  essential  points  with  Lobachevski's,  departing 
from  it  only  in  the  form  of  its  developments.  Ac- 
cording to  the  originals  which  have  been  made  al- 
most completely  accessible  to  us  in  the  beautiful 
editions  of  Engel  and  Staeckel,1  it  is  permissible  to 

1  Urlcunden  eur  GescMchte  der  nichteuklidischen  Geometric. 
L.  N.  I.  Lobatschefskij.    Leipzig,  1899. 


126 


SPACE    AND    GEOMETRY 


assume  that  Lobachevski  also  undertook  his  inves- 
tigations in  the  hope  of  becoming  involved  in  con- 
tradictions by  the  rejection  of  the  Euclidean  axiom. 
But  after  he  found  himself  mistaken  in  this  expec- 
tation, he  had  the  intellectual  courage  to  draw  all 
the  consequences  from  this  fact.  Lobachevski  gives 
his  conclusions  in  synthetic  form.  But  we  can 
fairly  well  imagine  the  general  analyzing  considera- 
tions that  paved  the  way  for  the  construction  of  his 
geometry. 

From  a  point  lying  outside  a  straight  line  g  ( Fig. 
23)  a  perpendicular  p  is  dropped  and  through  the 


Fig.  23. 

same  point  in  the  plane  pg  a  straight  line  h  is  drawn, 
making  with  the  perpendicular  an  acute  angle  s. 
Making  tentatively  the  assumption  that  g  and  h  do 
not  meet  but  that  on  the  slightest  diminution  of  the 
angle  s  they  would  meet,  we  are  at  once  forced  by 
the  homogeneity  of  space  to  the  conclusion  that  a 
second  line  k  having  the  same  angle  s  similarly  de- 
ports itself  on  the  other  side  of  the  perpendicular. 
Hence  all  non-intersecting  lines  drawn  through  the 
same  point  are  situate  between  h  and  k.  The  latter 
form  the  boundaries  between  the  intersecting  and 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  127 

non-intersecting  lines  and  are  called  by  Lobachev- 
ski  parallels. 

In  the  Introduction  to  his  New  Elements  of 
Geometry  (1835)  Lobachevski  proves  himself  a 
thorough  natural  inquirer.  No  one  would  think  of 
attributing  even  to  an  ordinary  man  of  sense  the 
crude  view  that  the  "parallel-angle"  was  very  much 
less  than  a  right  angle,  when  on  slight  prolongation 
it  could  be  distinctly  seen  that  they  would  intersect. 
The  relations  here  considered  admit  of  representa- 
tion only  in  drawings  that  distort  the  true  propor- 
tions, and  we  have  rather  to  picture  to  ourselves 
that  in  the  dimensions  of  the  illustration  the  vari- 
ation of  5"  from  a  right  angle  is  so  small  that  h 
and  k  are  to  the  eye  undistinguishably  coincident. 
Prolonging,  now,  the  perpendicular  p  to  a  point  be- 
yond its  intersection  with  h,  and  drawing  through 
its  extremity  a  new  line  /  parallel  to  h  and  therefore 
parallel  also  to  g,  it  follows  that  the  parallel-angle 
/  must  necessarily  be  less  than  s,  if  h  and  /  are  not 
again  to  fulfill  the  conditions  of  the  Euclidean  case. 
Continuing  in  the  same  manner,  the  prolongation  of 
the  perpendicular  and  the  drawing  of  parallels,  we 
obtain  a  parallel-angle  that  constantly  decreases. 
Considering,  now,  parallels  which  are  more  remote 
and  consequently  converge  more  rapidly  on  the  side 
of  convergence,  we  shall  logically  be  compelled  to 
assume,  not  to  be  at  variance  with  the  preceding 
supposition,  that  on  approach  or  on  the  decrease  of 
the  length  of  the  perpendicular  the  parallel-angle 
will  again  increase.  The  angle  of  parallelism, 


128 


SPACE    AND    GEOMETRY 


therefore,  is  an  inverse  function  of  the  perpendicu- 
lar p,  and  has  been  designated  by  Lobachevski  by 
n  (/>).  A  group  of  parallels  in  a  plane  has  the  ar- 
rangement shown  schematically  in  Figure  24.  They 
all  approach  one  another  asymptotically  toward  the 
side  of  their  convergence.  The  homogeneity  of 
space  requires  that  every  "strip"  between  two  paral- 
lels can  be  made  to  coincide  with  every  other  strip 
provided  it  be  displaced  the  requisite  distance  in  a 
longitudinal  direction. 


Fig.   24. 

If  a  circle  be  imagined  to  increase  indefinitely,  its 
radii  will  cease  to  intersect  the  moment  the  increas- 
ing arcs  reach  the  point  where  the  convergence  of 
the  radii  corresponds  to  parallelism.  The  circle  then 
passes  over  into  the  so-called  "boundary-line."  Sim- 
ilarly the  surface  of  a  sphere,  if  it  indefinitely  in- 
crease, will  pass  into  what  Lobachevski  calls  a 
"boundary-surface."  The  boundary-lines  bear  a 
relation  to  the  boundary-surface  analogous  to  that 
which  a  great  circle  bears  to  the  surface  of  a  sphere. 
The  geometry  of  the  surface  of  a  sphere  is  inde- 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS 


129 


pendent  of  the  axiom  of  parallels.  But  since  it  can 
be  demonstrated  that  triangles  formed  from  boun- 
dary-lines on  a  boundary-surface  no  more  exhibit 
an  excess  of  angle-sum  than  do  finite  triangles 
on  a  sphere  of  infinite  radius,  consequently  the 
rules  of  the  Euclidean  geometry  likewise  hold 
good  for  these  boundary-triangles.  To  find  points 
of  the  boundary-line,  we  determine  (Fig.  25) 

in    a    bundle    of    parallels,    aa,    bf3,    cy,    d8 

lying  in  a  plane  points  a,  b,  c,  d  in  each  of  these  par- 
allels so  situated  with  respect  to  the  point  a  in  aa. 


Fig.  25. 


that    L  aab  =  L  ftba,    L  aac  =  L  yea,       L   aad  = 

L  8da Owing  to   the   sameness   of   the   entire 

construction,  each  of  the  parallels  may  be  regarded 
as  the  "axis?'  of  the  boundary  line,  which  will  gen- 
erate, when  revolved  about  this  axis,  the  boundary- 
surface.  Likewise  each  of  the  parallels  may  be  re- 
garded as  the  axis  of  the  boundary-surface.  For 
the  same  reason  all  boundary-lines  and  all  boundary- 
surfaces  are  congruent.  The  intersection  of  every 
plane  with  the  boundary-surface  is  a  circle;  it  is  a 
boundary-line  only  when  the  cutting  plane  contains 


130  SPACE    AND    GEOMETRY 

the  axis.  In  the  Euclidean  geometry  there  is  no 
boundary-line,  nor  boundary-surface.  The  analo- 
gues of  them  are  here  the  straight  line  and  the  plane. 
If  no  boundary-line  exists,  then  necessarily  must 
any  three  points  not  in  a  straight  line  lie  on  a  circle. 
Hence  the  younger  Bolyai  was  able  to  replace  the 
Euclidean  axiom  by  this  last  postulate. 

Let  aa,  bft,  cy  be  a  system  of  parallels,  and  ae, 
fli^i,  a2e2.  .a  system  of  boundary-lines,  each  of  which 
systems  divides  the  other  into  equal  parts  (Fig.  25). 
The  ratio  to  each  other  of  any  two  boundary-arcs 
between  the  same  parallels,  e.  g.,  the  arcs  ae  =  u 
and  a2e2  =  u',  is  dependent  therefore  solely  on  their 
distance  apart  aa2  —  x.  We  may  put  generally 

u        * 

—?  =  e*,  where  k  is  so  chosen  that  e  shall  be  the 

base  of  the  Naperian  system  of  logarithms.  In  this 
manner  exponentials  and  by  means  of  these  hyper- 
bolic functions  are  introduced.  For  the  angle  of 

p 
parallelism   we    obtain   s=    cotyn(p)  =e  * .      If 

p  =  o,  *  =  -%',  if  p=  00,5  =  0. 

An  example  will  illustrate  the  relation  of  the  Lo- 
bachevskian  to  the  Euclidean  and  spherical  geom- 
etries. For  a  rectilinear  Lobachevskian  triangle 
having  the  sides  a,  b,  c,  and  the  angles  A,  B,  C,  we 
obtain,  when  C  is  a  right  angle, 

sinh  -=  sinh-sin  A. 

H  n> 

Here  sinh  stands  for  the  hyperbolic  sine, 

.  , 
sinh  x  — 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS 

whereas 

2J 

x   ,   xs      x*      x1 

or.  8mh*==ri+5i   JT   71 • 

x       x3   .   x6       x1   . 

and  sin  x  =  — ; r  -* — r r  + 

i  !      3 1     5!      7* 

Considering  the  relations  sin(^')  =i  (sinn*),  or 
sinh  (xi)  =  i  sin  x,  involved  in  the  foregoing  form- 
ulae, it  will  be  seen  that  the  above-given  formula  for 
the  Lobachevskian  triangle  passes  over  into  the 

formula  holding  for  the  spherical  triangle,  viz.,  siri  -j 
=  sin  ~  sin  A,  when  ki  is  put  in  the  place  of  k  in 

n 

the  former  and  k  is  considered  as  the  radius  of  the 
sphere,  which  in  the  usual  formulae  assumes  the 
value  unity.  The  re-transformation  of  the  spherical 
formula  into  the  Lobachevskian  by  the  same  method 
is  obvious.  If  k  be  very  great  in  comparison  with 
a  and  c,  we  may  restrict  ourselves  to  the  first  mem- 
ber of  the  series  for  sinh  or  sin,  obtaining  in  both 

cases,  -JT  =  4-  sin  A  or  a  =  csin^,  the  formula  of 
plane  Euclidean  geometry,  which  we  may  regard  as 
a  limiting  case  of  both  the  Lobachevskian  and  spher- 
ical geometries  for  very  large  values  of  k,  or  for 
A=  oo.  It  is  likewise  permissible  to  say  that  all 
three  geometries  coincide  in  the  domain  of  the  infi- 
nitely small. 


1F.  Engel,  N.  I.  LobatschefsMj,  Zwei  geometrische  Abhand- 
bmgen,  Leipsic,  1899. 


132  SPACE    AND    GEOMETRY 

THE  DIFFERENT  SYSTEMS  OF  GEOMETRY. 

As  we  see,  it  is  possible  to  construct  a  self-consist- 
ent, non-contradictory  system  of  geometry  solely  on 
the  assumption  of  the  convergence  of  parallel  lines. 
True,  there  is  not  a  single  observation  of  the  geomet- 
rical facts  accessible  to  us  that  speaks  in  favor  of 
this  assumption,  and  admittedly  the  hypothesis  is  at 
so  great  variance  with  our  geometrical  instinct  as 
easily  to  explain  the  attitude  toward  it  of  the  earlier 
inquirers  like  Saccheri  and  Lambert.  Our  imagina- 
tion, dominated  as  it  is  by  our  modes  of  visualizing 
and  by  the  familiar  Euclidean  concepts,  is  competent 
to  grasp  only  piecemeal  and  gradually  Lobachev- 
ski's  views.  We  must  suffer  ourselves  to  be  led  here 
rather  by  mathematical  concepts  than  by  sensuous 
images  derived  from  a  single  narrow  portion  of 
space.  But  we  must  grant,  nevertheless,  that  the 
quantitative  mathematical  concepts  by  which  we 
through  our  own  initiative  and  within  a  certain  arbi- 
trary scope  represent  the  facts  of  geometrical  expe- 
rience, do  not  reproduce  the  latter  with  absolute  ex- 
actitude. Different  ideas  can  express  the  facts  with 
the  same  exactness  in  the  domain  accessible  to  ob- 
servation. The  facts  must  hence  be  carefully  dis- 
tinguished from  the  intellectual  constructs  the  for- 
mation of  which  they  suggested.  The  latter — con- 
cepts— must  be  consistent  with  observation,  and 
must  in  addition  be  logically  in  accord  with  one  an- 
other. Now  these  two  requirements  can  be  fulfilled 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  133 

in  more  than  one  manner,  and  hence  the  different 
systems  of  geometry. 

Manifestly  the  labors  of  Lobachevski  were  the 
outcome  of  intense  and  protracted  mental  effort, 
and  it  may  be  surmised  that  he  first  gained  a  clear 
conception  of  his  system  from  general  considera- 
tions and  by  analytic  (algebraic)  methods  before  he 
was  able  to  present  it  synthetically.  Expositions  in 
this  cumbersome  Euclidean  form  are  by  no  means 
alluring,  and  it  is  possibly  due  mainly  to  this  fact 
that  the  significance  of  Lobachevski's  and  Bolyai's 
labors  received  such  tardy  recognition. 

Lobachevski  developed  only  the  consequences  of 
the  modification  of  Euclid's  Fifth  Postulate.  But  if 
we  abandon  the  Euclidean  assertion  that  "two 
straight  lines  cannot  enclose  a  space,"  we  shall  ob- 
tain a  companion-piece  to  the  Lobachevskian  geom- 
etry. Restricted  to  a  surface,  it  is  the  geometry  of 
the  surface  of  a  sphere.  In  place  of  the  Euclidean 
straight  lines  we  have  great  circles,  all  of  which 
intersect  twice  and  of  which  each  pair  encloses  two 
spherical  lunes.  There  are  therefore  no  parallels. 
Riemann  first  intimated  the  possibility  of  an  analo- 
gous geometry  for  three-dimensional  space  (of 
positive  curvature), — a  conception  that  does  not  ap- 
pear to  have  occurred  even  to  Gauss,  possibly  owing 
to  his  predilection  for  infinity.  And  Helmholtz,1 
who  continued  the  researches  of  Riemann  physically, 
neglected  in  his  turn,  in  his  first  publication,  the  de- 

"'Ueber    die   thatsftchlichen    Gnindlagen    der   Geometric," 
Wissensch.  Abhandl,  1866.    II.,  p.  610  et  seq. 


134  SPACE    AND    GEOMETRY 

velopment  of  the  Lobachevskian  case  of  a  space  of 
negative  curvature  (with  an  imaginary  parameter 
k).  The  consideration  of  this  case  is  in  point  of 
fact  more  obvious  to  the  mathematician  than  it  is  to 
the  physicist.  Helmholtz  treats  in  the  publication 
mentioned  only  the  Euclidean  case  of  the  curvature 
zero  and  Riemann's  space  of  positive  curvature. 

APPLICABILITY  OF  THE  DIFFERENT   SYSTEMS  TO 
REALITY. 

We  are  able,  accordingly,  to  represent  the  facts 
of  spatial  observation  with  all  possible  precision  by 
both  the  Euclidean  geometry  and  the  geometries  of 
Lobachevski  and  Riemann,  provided  in  the  two  lat- 
ter cases  we  take  the  parameter  k  large  enough. 
Physicists  have  as  yet  found  no  reason  for  depart- 
ing from  the  assumption  k  =  oo  of  the  Euclidean 
geometry.  It  has  been  their  practice,  the  result  of 
long  and  tried  experience,  to  adhere  steadfastly  to 
the  simplest  assumptions  until  the  facts  forced  their 
complication  or  modification.  This  accords  likewise 
with  the  attitude  of  all  great  mathematicians  to- 
ward applied  geometry.  The  deportment  of  phys- 
icists and  mathematicians  toward  these  ques- 
tions is  in  the  main  different,  but  this  is 
explained  by  the  circumstance  that  for  the 
former  class  of  inquirers  the  physical  facts  are  of 
most  significance,  geometry  being  for  them  merely 
a  convenient  implement  of  investigation,  while  for 
the  latter  class  these  very  questions  are  the  main 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  135 

material  of  research,  and  of  greatest  technical  and 
particularly  epistemological  interest.  Supposing  a 
mathematician  to  have  modified  tentatively  the  sim- 
plest and  most  immediate  assumptions  of  our  geo- 
metrical experience,  and  supposing  his  attempt  to 
have  been  productive  of  fresh  insight,  certainly 
nothing  is  more  natural  than  that  these  researches 
should  be  prosecuted  farther  from  a  purely  mathe- 
matical interest.  Analogues  of  the  geometry  we 
are  familiar  with,  are  constructed  on  broader  and 
more  general  assumptions  for  any  number  of  di- 
mensions, with  no  pretension  of  being  regarded  as 
more  than  intellectual  scientific  experiments  and 
with  no  idea  of  being  applied  to  reality.  In  sup- 
port of  my  remark  it  will  be  sufficient  to  advert  to 
the  advances  made  in  mathematics  by  Clifford, 
Klein,  Lie,  and  others.  Seldom  have  thinkers  be- 
come so  absorbed  in  revery,  or  so  far  estranged  from 
reality,  as  to  imagine  for  our  space  a  number  of 
dimensions  exceeding  the  three  of  the  given  space 
of  sense,  or  to  conceive  of  representing  that  space 
by  any  geometry  that  departs  appreciably  from  the 
Euclidean.  Gauss,  Lobachevski,  Bolyai,  and  Rie- 
mann  were  perfectly  clear  on  this  point,  and  cannot 
certainly  be  held  responsible  for  the  grotesque  fic- 
tions which  were  subsequently  constructed  in  this 
domain. 

It  little  accords  with  the  principles  of  a  physicist 
to  make  suppositions  regarding  the  deportment  of 
geometrical  constructs  in  infinity  and  in  non-acces- 
sible places,  then  subsequently  to  compare  them 


136  SPACE    AND    GEOMETRY 

with  our  immediate  experience  and  adapt  them  to 
it.  He  prefers,  like  Stolz,  to  regard  what  is  directly 
given  as  the  source  of  his  ideas,  which  he  likewise 
considers  applicable  to  what  is  inaccessible  until 
obliged  to  change  them.  But  he  too  may  be  ex- 
tremely grateful  for  the  discovery  that  there  exist 
several  sufficing  geometries,  that  we  can  make  shift 
also  with  a  Unite  space,  etc., — grateful  in  short,  for 
the  abolition  of  certain  conventional  barriers  of 
thought. 

If  we  lived  on  the  surface  of  a  planet  with  a  tur- 
bid, opaque  atmosphere  and  if,  on  the  supposition 
that  the  surface  of  the  earth  was  a  plane  and  our 
only  instruments  were  square  and  chain,  we  were 
to  undertake  geodetic  measurements;  then  the  in- 
crease in  the  excess  of  the  angle-sum  of  large  tri- 
angles would  soon  compel  us  to  substitute  a  spher- 
ometry  for  our  planimetry.  The  possibility  of  an- 
alogous experiences  in  three-dimensional  space  the 
physicist  cannot  as  a  matter  of  principle  reject,  al- 
though the  phenomena  that  would  compel  the  ac- 
ceptance of  a  Lobachevskian  or  a  Riemannian  ge- 
ometry would  present  so  odd  a  contrast  with  those 
to  which  we  have  been  hitherto  accustomed,  that  no 
one  will  regard  their  actual  occurrence  as  probable. 

The  question  whether  a  given  physical  object  is 
a  straight  line  or  the  arc  of  a  circle  is  not  properly 
formulated.  A  stretched  chord  or  a  ray  of  light  is 
certainly  neither  the  one  nor  the  other.  The  ques- 
tion is  simply  whether  the  object  so  spatially  reacts 
that  it  conforms  better  to  the  one  concept  than  to 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  137 

the  other,  and  whether  with  the  exactitude  which  is 
sufficient  for  us  and  obtainable  by  us  it  conforms  at 
all  to  any  geometric  concept.  Excluding  the  latter 
case,  the  question  arises,  whether  we  can  in  practice 
remove,  or  at  least  in  thought  determine  and  make 
allowance  for,  the  deviation  from  the  straight  line 
or  circle,  in  other  words,  correct  the  result  of  the 
measurement.  But  we  are  dependent  always,  in 
practical  measurements,  on  the  comparison  of  phys- 
ical objects.  If  on  direct  investigation  these  coin- 
cided with  the  geometric  concepts  to  the  highest  at- 
tainable point  of  accuracy,  but  the  indirect  results 
of  the  measurement  deviated  more  from  the  theory 
than  the  consideration  of  all  possible  errors  per- 
mitted, then  certainly  we  should  be  obliged  to 
change  our  physico-metric  notions.  The  physicist 
will  do  well  to  await  the  occurrence  of  such  a  situa- 
tion, while  in  the  meantime  the  mathematician  may 
be  allowed  full  and  free  scope  for  his  speculations. 

THE  CONCEPTS  OF  MATHEMATICS  AND  PHYSICS. 

Of  all  the  concepts  which  the  natural  inquirer 
employs,  the  simplest  are  the  concepts  of  space  and 
time.  Spatial  and  temporal  objects  conforming  to 
his  conceptual  constructs  can  be  framed  with  great 
exactness.  Nearly  every  observable  deviation  can 
be  eliminated.  We  can  imagine  any  spatial  or  tem- 
poral construct  realized  without  doing  violence  to 
any  fact.  The  other  physical  properties  of  bodies 
are  so  intimately  interconnected  that  in  their  case 
arbitrary  fictions  are  subjected  to  narrow  restric- 


138  SPACE    AND    GEOMETRY 

tions  by  the  facts.  A  perfect  gas,  a  perfect  fluid,  a 
perfectly  elastic  body  does  not  exist;  the  physicist 
knows  that  his  fictions  conform  only  approximately 
and  by  arbitrary  simplifications  to  the  facts;  he  is 
perfectly  aware  of  the  deviation,  which  cannot  be  re- 
moved. We  can  conceive  a  sphere,  a  plane,  etc., 
constructed  with  unlimited  exactness,  without  run- 
ning counter  to  any  fact.  Hence,  when  any  new 
physical  fact  occurs  which  renders  a  modification  of 
our  concepts  necessary,  the  physicist  always  prefers 
to  sacrifice  the  less  perfect  concepts  of  physics  rather 
than  the  simpler,  more  perfect,  and  more  lasting 
concepts  of  geometry,  which  form  the  solidest 
foundation  of  all  his  theories. 

But  the  physicist  can  derive  in  another  direction 
substantial  assistance  from  the  labors  of  geometers. 
Our  geometry  refers  always  to  objects  of  sensuous 
experience.  But  the  moment  we  begin  to  operate 
with  mere  things  of  thought  like  atoms  and  mole- 
cules, which  from  their  very  nature  can  never  be 
made  the  objects  of  sensuous  contemplation,  we  are 
under  no  obligation  whatever  to  think  of  them  as 
standing  in  spatial  relationships  which  are  peculiar 
to  the  Euclidean  three-dimensional  space  of  our  sen- 
suous experience.  This  may  be  recommended  to  the 
special  attention  of  thinkers  who  deem  atomistic 
speculations  indispensable.1 


1  While  still  an  upholder  of  the  atomic  theory,  I  sought  to 
explain  the  line-spectra  of  gases  by  the  vibrations  of  the  atomic 
constituents  of  a  gas-molecule  with  respect  to  another.  The 
difficulties  which  I  here  encountered  suggested  to  me  (1863) 
the  idea  that  non-sensuous  things  did  not  necessarily  have  to 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS         139 

THE  RELATIVITY  OF  ALL  SPATIAL  RELATIONS. 

Let  us  go  back  in  thought  to  the  origin  of  geom- 
etry in  the  practical  needs  of  life.  The  recognition 
of  the  spatial  substantiality  and  spatial  invariability 
of  spatial  objects  in  spite  of  their  movements  is  a 
biological  necessity  for  human  beings,  for  spatial 
quantity  is  related  directly  to  the  quantitative  satis- 
faction of  our  needs.  When  knowledge  of  this  sort 
is  not  sufficiently  provided  for  by  our  physiological 
organization,  we  employ  our  hands  and  feet  for 
comparing  the  spatial  objects.  When  we  begin  to 
compare  bodies  with  one  another,  we  enter  the 
domain  of  physics,  whether  we  employ  our  hands 
or  an  artificial  measure.  All  physical. determinations 
are  relative.  Consequently,  likewise  all  geomet- 
rical determinations  possess  validity  only  relatively 
to  the  measure.  The  concept  of  measurement  is 
a  concept  of  relation,  which  contains  nothing  not 
contained  in  the  measure.  In  geometry  we  sim- 
ply assume  that  the  measure  will  always  and  every- 
where coincide  with  that  with  which  it  has  at 
some  other  time  and  in  some  other  place  coincided. 
But  this  assumption  is  determinative  of  nothing  con- 
be  pictured  in  our  sensuous  space  of  three  dimensions.  In  this 
way  I  also  lighted  upon  analogues  of  spaces  of  different  num- 
bers of  dimensions.  The  collateral  study  of  various  physio- 
logical manifolds  (see  footnote  on  page  98  of  this  book)  led 
me  to  the  problems  discussed  in  the  conclusion  of  this  paper. 
The  notion  of  finite  spaces,  converging  parallels,  etc.,  which 
can  come  only  from  a  historical  study  of  geometry,  was  at  that 
time  remote  from  me.  I  believe  that  my  critics  would  have 
done  well  had  they  not  overlooked  the  italicised  paragraph. 
For  details  see  the  notes  to  my  Erhaltvng  der  Arbeit,  Prague, 
1872. 


I4O  SPACE    AND    GEOMETRY 

cerning  the  measure.  In  place  of  spatial  physiolog- 
ical equality  is  substituted  an  altogether  differently 
defined  physical  equality,  which  must  not  be  con- 
founded with  the  former,  no  more  than  the  indica- 
tions of  a  thermometer  are  to  be  identified  with  the 
sensation  of  heat.  The  practical  geometer,  it  is  true, 
determines  the  dilatation  of  a  heated  measure,  by 
means  of  a  measure  kept  at  a  constant  temperature, 
and  takes  account  of  the  fact  that  the  relation  of  con- 
gruence in  question  is  disturbed  by  this  non-spatial 
physical  circumstance.  But  to  the  pure  theory  of 
space  all  assumptions  regarding  the  measure  are  for- 
eign. Simply  the  physiologically  created  habit  of 
regarding  the  measure  as  invariable  is  tacitly  but  un- 
justifiably retained.  It  would  be  quite  superfluous 
and  meaningless  to  assume  that  the  measure,  and 
therefore  bodies  generally,  suffered  alterations  on 
displacement  in  space,  or  that  they  remained  un- 
changed on  such  displacement, — a  fact  which  in 
its  turn  could  only  be  determined  by  the  use  of  a 
new  measure.  The  relativity  of  all  spatial  relations 
is  made  manifest  by  these  considerations. 

INTRODUCTION  OF  THE  NOTION  OF  NUMBER. 

If  the  criterion  of  spatial  equality  is  substantially 
modified  by  the  introduction  of  measure,  it  is  sub- 
jected to  a  still  further  modification,  or  intensifica- 
tion, by  the  introduction  of  the  notion  of  number 
into  geometry.  There  is  nicety  of  distinction  gained 
by  this  introduction  which  the  idea  of  congruence 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  14! 

alone  could  never  have  attained.  The  application 
of  arithmetic  to  geometry  leads  to  the  notion  of  in- 
commensurability and  irrationality.  Our  geometric 
concepts  therefore  contain  adscititious  elements  not 
intrinsic  to  space;  they  represent  space  with  a  cer- 
tain latitude,  and,  arbitrarily  also,  with  greater  pre- 
cision than  spatial  observation  alone  could  possibly 
ever  realize.  This  imperfect  contact  between  fact 
and  concept  explains  the  possibility  of  different  sys- 
tems of  geometry.1 

SIGNIFICANCE  OF  THE  METAGEOMETRIC  MOVE- 
MENT. 

The  entire  movement  which  led  to  the  transforma- 
tion of  our  ideas  of  geometry  must  be  characterized 
as  a  sound  and  healthful  one.  This  movement, 
which  began  centuries  ago  but  is  now  greatly  inten- 
sified, is  not  to  be  looked  upon  as  having  terminated. 
On  the  contrary,  we  are  quite  justified  in  the  ex- 
pectation that  it  will  long  continue,  and  redound  not 
only  to  the  great  advancement  of  mathematics  and 
geometry,  especially  in  an  epistemological  regard, 
but  also  to  that  of  the  other  sciences.  This  move- 
ment was,  it  is  true,  powerfully  stimulated  by  a  few 
eminent  men,  but  it  sprang,  nevertheless,  not  from 
an  individual,  but  from  a  general  need.  This 
will  be  seen  from  the  difference  in  the  pro- 

*It  would  be  too  much  to  expect  of  matter  that  it  should 
realize  all  the  atomistic  fantasies  of  the  physicist.  So,  too, 
space,  as  an  object  of  experience,  can  hardly  be  expected  to 
satisfy  all  the  ideas  of  the  mathematician,  though  there  be  no 
doubt  whatever  as  to  the  general  value  of  their  investigations. 


142  SPACE    AND    GEOMETRY 

fessions  of  the  men  who  have  taken  part  in  it.  Not 
only  the  mathematician,  but  also  the  philosopher 
and  the  educationist,  have  made  considerable  contri- 
butions to  it.  So,  too,  the  methods  pursued  by  the 
different  inquirers  are  not  unrelated.  Ideas  which 
Leibnitz1  uttered  recur  in  slightly  altered  form  in 
Fourier,2  Lobachevski,  Bolyai,  and  H.  Erb.3  The 
philosopher  Ueberweg,4  closely  approaching  in  his 
opposition  to  Kant  the  views  of  the  psychologist 
Beneke,5  and  in  his  geometrical  ideas  starting  from 
Erb  (which  later  writer  mentions  K.  A.  Erb8  as  his 
predecessor)  anticipates  a  goodly  portion  of  Helm- 
holtz's  labors. 

SUMMARY. 

The  results  to  which  the  preceding  discussion  has 
led,  may  be  summarized  as  follows: 

1.  The  source  of  our  geometric  concepts  has 
been  found  to  be  experience. 

2.  The  character  of  the  concepts  satisfying  the 


1  See  above  pp.  66-67. 

8  Stances  de  I'Ecole  Normale.    Debats.    Vol.  I.,  1800,  p.  28. 

"H.  Erb,  Grossherzoglich  Badischer  Finanzrath,  Die  Pro- 
bleme  der  geraden  Linie,  des  Wirikels  und  der  ebenen  FldcJie, 
Heidelberg,  1846. 

•"Die  Principien  der  Geometric  wissenschaftlich  darge- 
stellt."  Archvo  fur  Philologie  und  Pddagogik.  1851.  Be- 
printed  in  Brasch's  Welt-  und  Lebensanschauung  F.  Ueber- 
wegs,  Leipzig,  1889,  pp.  263-317. 

*Logilc  als  Kunstlehre  des  Derikens,  Berlin,  1842,  Vol.  II., 
pp.  51-55. 

•Zur  Mathematilc  und  Logilc,  Heidelberg,  1821.  I  was  un- 
able to  examine  this  work. 


FROM  THE  POINT  OF  VIEW  OF  PHYSICS  143 

same  geometrical  facts  has  been  shown  to  be  many 
and  varied. 

3.  By  the  comparison  of  space  with  other  mani- 
folds, more  general  concepts  have  been  reached,  of 
which  the  geometric  represents  a  special  case.    Geo- 
metric thought  has  thus  been  freed  from  conven- 
tional limitations,  heretofore  imagined  insuperable. 

4.  By  the   demonstration  of  the  existence   of 
manifolds  allied  to  but  different  from  space,  en- 
tirely new  questions  have  been  suggested.     What 
is  space  physiologically,  physically,  geometrically? 
To  what  are  its  specific  properties  to  be  attributed, 
since  others  are  also  conceivable?     Why  is  space 
three-dimensional,  etc.  ? 

With  questions  such  as  these,  though  we  must  not 
expect  the  answer  to-day  or  to-morrow,  we  stand 
before  the  entire  profundity  of  the  domain  to  be 
investigated.  We  shall  say  nothing  of  the  inept 
strictures  of  the  Boeotians,  whose  coming  Gauss 
predicted,  and  whose  attitude  determined  him  to  re- 
serve. But  what  shall  we  say  to  the  acrid  and  cap- 
tious criticisms  to  which  Gauss,  Riemann  and  their 
associates  have  been  subjected  by  men  of  highest 
standing  in  the  scientific  world?  Have  these  men 
never  experienced  in  their  own  persons  the  truth 
that  inquirers  on  the  outermost  boundaries  of 
knowledge  frequently  discover  many  things  that 
will  not  slip  smoothly  into  all  heads,  but  which  are 
not  on  that  account  arrant  nonsense?  True,  such 
inquirers  are  liable  to  error,  but  even  the  errors  of 
some  men  are  often  more  fruitful  in  their  conse- 
quences than  the  discoveries  of  others. 


INDEX. 


Actite  angle,  119. 
Agrimensores,  46. 
Angle,  no;  Measurement  of  the, 

76. 
Angle-sum   of   a  triangle,    116;    of 

large   triangles,    136. 
Anschauung,    61  n,  ff.,    75,   83,   89, 

90. 

Anti-Euclidean   geometry,    122. 
A  priori  theory  of  space,  34. 
Archytas,   37. 
Area,   Measures  of,   70. 
Aristarchus,    37. 
Atomic  theory,    138  n. 
Attention,    Locality   of  the,    13. 
Augustine,   St.,  37. 
Axioms,   Eleventh  and  Twelfth, 

114. 

Becker,    Bernard,    in   n. 

Biological  necessity  paramount  in 
the  perception  of  space,  25; 
theory  of  spatial  perception,  32. 

Blind  person,   Sensations  of,  20. 

Bodies,  All  measurement  by,  47; 
Geometry  requires  experience 
concerning,  38  ff.;  Spatial  per- 
manency of,  82;  Spatially  un- 
alterable physical,  71. 

Bolyai,    119,    125  ff. 

Boundaries,   126. 

Boyle's  law,   58. 

Breuer,   27  n. 

Cantor,   M.,   46  n.,    sin.,    72  n. 
Cartesian  geometry,  36;  system  ol 

co-ordinates,    83. 
Cavalieri,  50. 
Children,   Drawings  of,   55. 


Circular  meffsurf,  «IO. 

Clifford,   135. 

Color-sensations,   99. 

Colors,   Analogies  of  space  with, 

98. 

Comparison,    Bodies   of,    45. 
Concepts    of    geometry,    84;     arise 

on   the  basis   of   experience,    82. 
Constancy,  Notion  of,  40;  Spatial, 

53- 

Counting,    Experiments    in,    98    n. 
Curvature,    Measure   of,    107    ff. 

Deductive  geometry,    112. 

Delbceuf,   119. 

De  Tilly,  80  n. 

Directions,   Three  cardinal,    18. 

Distance,   78. 

Dvorak,  24. 

Egyptians,  Land-surveying  among, 
54- 

Eisenlohr,   46  n. 

Equality,  44. 

Erb,    142. 

Euclid,  63  n.,  113,  114,  116  ff.; 
Space  of,  6. 

Euclidean  geometry,  71  n.,  116  ff. 

Eudemus,    54. 

Euler,  7. 

Experience,  source  of  our  knowl- 
edge, 63. 

Experimenting  in  thought,  75,  86. 

Fictions,    138. 

Figures  in  demonstrations,  incor- 
rect, 93. 

Fluid,  A  perfect,  138. 
Fourier,   142. 


146 


SPACE  AND  GEOMETRY. 


Galileo,   100. 

Gas,  A  perfect,   138. 

Gauss,  50,  84,  97,   115,   121,   143. 

Geometric  concepts,  94;  and  phys- 
iological space,  Correspondence 
of,  it ;  instruction,  68;  knowl- 
edge, Various  sources  of  our, 
83;  space,  5  ff. 

Geometry,  compared  with  physics, 
124;  Applicability  of  the  differ- 
ent systems  of,  to  reality,  134; 
Concepts  of,  arise  on  the  basis 
of  experience,  82;  concerned 
with  ideal  objects  produced  by 
the  schematization  of  experi- 
mental objects,  68;  Deductive, 
112;  Early  discoveries  in,  109; 
Empirical  origin  of,  67,  109; 
Experimental,  58;  from  the 
point  of  view  of  physical  in- 
quiry, 94  ff. ;  Fundamental  as- 
sumptions of,  42,  49,  97;  Fun- 
damental facts  and  concepts  of, 
84;  Physical  origin  of,  43;  Phys- 
iological influences  in,  35;  Prac- 
tical origin  of,  53;  Present  form 
of,  90;  Psychology  and  natural 
development  of,  38  ff.;  Real 
problem  of,  70;  Riemann's  con- 
ception of,  96;  requires  expe- 
rience concerning  bodies,  38  ff. ; 
Role  of  volume  in  the  begin- 
nings of,  45;  Systems  of  94, 
132  ff.;  Technical  and  scientific 
development  of,  69 ;  Visual  sense 
in,  81. 

Gerhardt,    51   n. 

Giordano,   Vitale,   64  n. 

Hankel,   56  n. 

Haptic  space,  10.    See  also  Touch, 

Space  of. 

Helmholtz,    51,    133. 
Herbart,    113. 
Hering,   5,    13,    14,    19. 
Herodotus,  36,   47   n.,   54. 
Holder,    49  n. 

Idealizations  of  experience,  97. 
Imaginary   quantities,    104. 
Impenetrability,   73. 


Incommensurability,    141. 
Indivisibles,  Method  of,  51. 
Inequality,  44. 
Infinitesimal   method,    49. 
Irrationality,    141. 

James,   Prof.  William,   13,    14,    15. 

Kant,    34,    80,    90. 
Killing,   W.,   49  n. 
Klein,   135. 
Kosack,  in  n. 
Kronecker,   108. 

Labyrinthine  canals,  27. 

Lactantius,   36. 

Lagrange,   103. 

Lambert,    116,    120  ff. 

Land-surveying  among  the  Egyp- 
tians, 54. 

Leibnitz,  50,  64  n.,  66,  67  n.,  71  n. 

Length  as  fundamental  measure, 
78. 

Lever,  Law  of  the,   59. 

Lie,   135. 

Lines,    49. 

Lobachevski,     119;    and    Bolyai, 
Contributions  of   125   ff. 

Locality  of  sensation,  7. 

Locative   qualities,    95. 

Loeb,  23  n. 

Manifold,  Riemann's  conception 
of  an  n-fold  continuous,  105. 

Manifolds,  109;  Multiply-extend- 
ed, 98  ff. 

Mathematics    and   physics,    Con- 
cepts of,   137. 

Measure,  Concept  of,  71;  Funda- 
mental, 78,  no;  of  curvature, 
121. 

Measures  dependent  on  one  an- 
other, real  problem  of  geometry, 
70;  Names  of  the  oldest,  45. 

Measurement,  44;  a  physical  reac- 
tion, 62;  Volume  the  basis  of, 
81. 

Mental   experiment,    86. 

Metageometric  movement,  Signifi- 
cance of  the,  141. 

Metageometricians,    Space   of,    6. 


INDEX. 


147 


Metageametry,  94  n. 
Metric  space,  7. 

Motion,  Rate  of,  felt  directly,  24. 
Movement,    Sensations  of,   27 
Mullet,  Johannes,   5. 
Multiply  -  extended    "magnitude," 
97- 

Necessity,  Biological,  paramount 
in  the  perception  of  space,  25. 

Number,  Introduction  of  the  no- 
tion of,  140;  a  product  of  the 
mind,  98  n. 

Obtuse  angle,  119. 

Oppel,  24- 

Ornamental  themes,   55. 

Paper-folding,   57. 

Papyrus  Rhind,  46  n. 

Parabola,  Quadrature  of  the,  52. 

Parallels,  112,  114,  127;  Saccheri's 
theory  of,  116;  Theorem  of, 
59- 

Paving,  59. 

Physical  origin  of  geometry,  43, 
61. 

Physics  and  mathematics,  Concepts 
off  I37;  Properties  of  physio- 
logical space  traceable  in,  36. 

Physiological     and     geometrical 
space,   5   ff. 

Plane,  The,  64,  65. 

Plateau,   6,  24. 

Playfair,  56  n. 

Postulate,  Fifth,   114. 

Proclus,    54. 

Psychology  and  natural  develop- 
ment of  geometry,  38  ff 

Ptolemy,   7. 

Purkynje,   28. 

Pythagorean  theorem,  55,  72. 

Rays  of  light,  82. 

Relativity  of  all  spatial  relations, 

139- 
Riemann,    105;    his   conception   of 

geometry,   9,   96   ff. 
Rigidity,   Notion  of,  42. 
Rope-stretching,    82. 


Rotation,  109;  about  the  vertical 
axis,  24. 

Saccheri's    theory    of    parallels, 
116  ff. 

Saunderson,    blind   geometer,    21. 

Scholasticism,    114. 

Schopenhauer,   113. 

Schweickart,    125. 

Scientific  presentation,    113. 

Semicircle,  All  angles  right  angles 
in  a,  61. 

Sensation,  All,  spatial  in  charac- 
ter, 13;  in  its  biological  rela- 
tionship, 16;  of  movement,  27. 

Sensations,  Complexes  of,  39;  Iso- 
lated, 39. 

Sensational  qualities,  95. 

Sense-impression,    33. 

Sensible  space,  a  system  of  grad- 
uated feelings,  17. 

Skin,    Space-sense  of  the,   8. 

Snell,   70. 

Sophists,    1 13. 

Sosikles,  36. 

Sound,   Localizing  sources  of,    14. 

Space,  A  priori  theory  of,  34; 
Analogies  of,  with  colors,  98; 
Analogies  of,  with  time,  100; 
and  time  as  sensational  mani- 
folds, 102;  Biological  necessity 
paramount  in  the  perception  of, 
25;  Correspondence  of  physio- 
logical and  geometric,  n;  Feel- 
ings of,  involve  stimulus  to  mo- 
tion, 22;  Four-dimensional,  108; 
from  the  point  of  view  of  phys- 
ical inquiry,  94  ff. ;  Mere  pas- 
sive contemplation  of,  62;  Met- 
ric, 7;  of  Euclid,  6;  of  meta- 
geometricians,  6;  of  touch,  7ff. ; 
of  vision,  5  ff. ;  Our  notions  of, 
94;  Physiological  and  geomet- 
rical, 5;  Primary  and  second- 
ary, 30;  Riemannian,  9;  Sense 
of,  dependent  on  biological 
function,  10  if.;  Three-dimen- 
sional, 1 08. 

Space-sensation,  33;  System  of, 
finite,  1 1 . 

Spatial   perception,   Biological  the- 


148 


SPACE   AND   GEOMETRY. 


ory  of,  32;  relations,  Relativity 
of  all,  139. 

Steiner,   70. 

Steinhauser,    14. 

Stolz,   122,   136. 

Stone-workers  of  Assyria,  Egypt, 
Greece,  etc.,  55. 

Straight  line,  62,  63,  75,  86  n., 
no,  136. 

Substantiality,  Spatial,  41;  Tem- 
poral, 100. 

Surfaces,  49;  Measurement  of,  46, 
69;  of  constant  curvature,  107. 

Symbols,    108;    Exetension   of,  103. 

Symmetric   figures,    80. 

Tactual  and  visual  space  corre- 
lated, 19;  sensibility,  10;  space, 
Biological  importance  of,  18. 

Taurinus,    125. 

Teleological  explanation  of  sense- 
adaptation,  12. 

Thibaut,    56. 

Thought,    Experiment   in,    62,    86. 

Three  dimensions  of  space,  98  n.; 
Origin  of  the,  18. 

Thucydides,    47    n. 

Time  and  space  as  sensational 
manifolds,  Difference  of  the 
analogies  of,  102. 

Tonal  sensations,    101. 

Tones,    spatial,    14. 

Touch,    Space   of,    7    ff. 

Trendelenburg,    113, 


Triangle,    Angle-sum    of,    56,    59, 

in;   Geometry  of  the,   71. 
Troltsch,    14  n. 
Tropisms  of  animals,  62 
Tylor,   57. 

Ueberweg,    142. 

Validity,    Universal,    92. 

Vision,   Space  of,   5  ff. 

Visual  and  tactual  space  corre- 
lated, 19;  sense  in  geometry,  81. 

Visualization  (Anschauung) ,  61 
ff-,  75,  83,  89,  90;  of  space, 
qualitative,  87. 

Volume  of  bodies,  Conservation 
of  the,  73;  Role  of,  in  the  be- 
ginnings of  geometry,  45;  the 
basis  of  measurement,  81. 

Voluminousness  ascribed  to  sen- 
sations, 14. 

Wallis,    119. 
Weaving,    54. 
Weber,   E.   H.,   8,    10. 
Weissenborn,   51  n. 
Will,  The   central  motor  organ    and 
the,   to  move,   25. 

Xerxes,    47. 

Zindler,   91   n. 
Zoth,  7. 


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"Our  notions  of  space  are  rooted  in  our 

physiological  organism.  Geometric 

concepts  are  the  product  of  the 

idealization  of  physical  experiences  of 

space.  Systems  of  geometry,  finally, 

originate  in  the  logical  classification  of 

the  conceptual  materials  so  obtained.  All 

three  factors  have  left  their  indubitable 

traces  in  modern  geometry. 

Epistemological  inquiries  regarding 

space  and  geometry  accordingly 

concern, ijie  physiologist,  the 

psychologist,  the  phvsicist,  the 

mathematician,  the  philosopher,  and 

the  logician  alike,  and  thev  can  be 

gradually  carried  to  their  definitive 

solution  only  by  the  consideration  of  the 

widely  disparate  points  of  view  which  are 

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